{"id":"n-f680a3b70a20ba67","revision":0,"first_index":0,"fingerprint":"f6-f06e51c139cfae05","name":"derivative of basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[[],["15"],["180","270"],["1260","3780","2835"],["6720","30240","45360","22680"],["30240","181440","408240","408240","153090"],["120960","907200","2721600","4082400","3061800","918540"],["443520","3991680","14968800","29937600","33679800","20207880","5051970"],["1520640","15966720","71850240","179625600","269438400","242494560","121247280","25981560"],["4942080","59304960","311351040","934053120","1751349600","2101619520","1576214640","675520560","126660105"],["15375360","207567360","1245404160","4358914560","9807557760","14711336640","14711336640","9457287840","3546482940","591080490"],["46126080","691891200","4670265600","18681062400","49037788800","88268019840","110335024800","94572878400","53197244100","17732414700","2659862205"],["134184960","2214051840","16605388800","74724249600","224172748800","470762772480","706144158720","756583027200","567437270400","283718635200","85115590560","11606671440"],["380190720","6843432960","56458321920","282291609600","952734182400","2286562037760","4001483566080","5144764584960","4823216798400","3215477865600","1446965039520","394626828960","49328353620"],["1052835840","20530298880","184772689920","1016249794560","3810936729600","10289529169920","20579058339840","30868587509760","34727160948480","28939300790400","17363580474240","7103282921280","1775820730320","204902391960"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","15","450","7875","105000","1181250","11812500","108281250","928125000","7541015625","58652343750","439892578125","3199218750000","22661132812500","156884765625000"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["15"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","12"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185923,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-b66d4f3c62f1524f"]},{"id":185924,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-b66d4f3c62f1524f"]},{"id":185925,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-b66d4f3c62f1524f"]},{"id":185926,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-b66d4f3c62f1524f"]}],"region_exact":{},"incoming_edges":[{"id":"e-b66d4f3c62f1524f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-f680a3b70a20ba67","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"d7a8c4c50ff60b1f0c94dd74c00bda0bb8fb074bb8177a86bb312e84f777395d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-53e421fb0a6e1da4","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-010e1119218a93c8","kind":"fingerprint_related","node_a_id":"n-f6495b2c898daf02","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-f6495b2c898daf02","other":{"id":"n-f6495b2c898daf02","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","6"],["27","36","12"]],"card_rows":[["3"],["9","6"],["27","36","12"],["81","162","108","24"]],"row_sums_preview":["3","15","75","375","1875","9375","46875","234375"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013609"],"edges":["e-618e7f7547136f14"],"text":"A120910 \u2192 basis A013609"},"headline":"A120910 \u2192 basis A013609","chain":null}},{"id":"l-0123f4baf7f69ce9","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-daeeef1566d61733","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-daeeef1566d61733","other":{"id":"n-daeeef1566d61733","name":"basis_map_A013609 of shift_x_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","6"],["54","72","24"]],"card_rows":[["1"],["9","6"],["54","72","24"],["270","540","360","80"]],"row_sums_preview":["1","15","150","1250","9375","65625","437500","2812500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-34e3d9a08d46fb54","e-3183bd8d49ca6dec"],"text":"A094305 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A094305 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-0363ac7dd9b0c13b","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-f212d5b113b473a6","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-f212d5b113b473a6","other":{"id":"n-f212d5b113b473a6","name":"basis_map_A013610 of basis_map_A013609 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","24"],["160","480","360"]],"card_rows":[["1"],["16","24"],["160","480","360"],["1280","5760","8640","4320"]],"row_sums_preview":["1","40","1000","20000","350000","5600000","84000000","1200000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013609","basis A013610"],"edges":["e-9fe823de1e8617e8","e-2087dcd07d7f483a"],"text":"A178820 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A178820 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-0777f00575e35f92","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-ca6d44114ccba1b3","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-ca6d44114ccba1b3","other":{"id":"n-ca6d44114ccba1b3","name":"basis_map_A013609 of basis_map_A013610 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-6"],["16","48","36"]],"card_rows":[["1"],["-4","-6"],["16","48","36"],["-64","-288","-432","-216"]],"row_sums_preview":["1","-10","100","-1000","10000","-100000","1000000","-10000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013610","basis A013609"],"edges":["e-226103a223be7e17","e-bf97e76c24088272"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013610 \u2192 basis A013609"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-086a28e19a7e7941","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-84c6dce3a7ed45cc","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-84c6dce3a7ed45cc","other":{"id":"n-84c6dce3a7ed45cc","name":"basis_map_A013610 of Triangle T(n,k) = C(n+k-1,k)*C(2*n-1,n-k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["12","36","27"]],"card_rows":[["1"],["2","3"],["12","36","27"],["80","360","540","270"]],"row_sums_preview":["1","5","75","1250","21875","393750","7218750","134062500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9a3e4fcd8c6a0286","steps":["basis A013610"],"edges":["e-590ce28009a1ef68"],"text":"A258758 \u2192 basis A013610"},"headline":"A258758 \u2192 basis A013610","chain":null}},{"id":"l-0a8e3128ced6aca7","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-819b1fbbf31fdfda","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-819b1fbbf31fdfda","other":{"id":"n-819b1fbbf31fdfda","name":"derivative of basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["18"],["252","378"]],"card_rows":[["18"],["252","378"],["2016","6048","4536"],["12096","54432","81648","40824"]],"row_sums_preview":["0","18","630","12600","189000","2362500","25987500","259875000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013610","D"],"edges":["e-3fa86ece9d0bedd1","e-fc2addf2dff3d3ba"],"text":"A178822 \u2192 basis A013610 \u2192 D"},"headline":"A178822 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-0ba8eac93d823922","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-d16ab8907dc7093c","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-d16ab8907dc7093c","other":{"id":"n-d16ab8907dc7093c","name":"reversal of basis_map_A013610 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["18","24","8"]],"card_rows":[["1"],["3","2"],["18","24","8"],["135","270","180","40"]],"row_sums_preview":["1","5","50","625","8750","131250","2062500","33515625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013610","reverse"],"edges":["e-13e67b7b83b67a66","e-49b7ec4756dd8075"],"text":"A085880 \u2192 basis A013610 \u2192 reverse"},"headline":"A085880 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-0d7e33fbcb02b2a3","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-bc1ce0156476df16","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-bc1ce0156476df16","other":{"id":"n-bc1ce0156476df16","name":"basis_map_A013609 of shift_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","8"],["81","108","36"]],"card_rows":[["1"],["12","8"],["81","108","36"],["432","864","576","128"]],"row_sums_preview":["1","20","225","2000","15625","112500","765625","5000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-18ace392e620d7bd","e-98669a687f79357d"],"text":"A135065 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A135065 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-0de01b7fe5205260","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-2c4d4128b8594c17","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-2c4d4128b8594c17","other":{"id":"n-2c4d4128b8594c17","name":"basis_map_A013609 of shift_x_plus_one of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["6","4"]],"card_rows":[["1"],["6","4"],["27","36","12"],["108","216","144","32"]],"row_sums_preview":["0","1","10","75","500","3125","18750","109375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-1a916fd0c60fb95f","e-c329575e2524f635"],"text":"A003506 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A003506 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-113e99fe121ab8ac","kind":"fingerprint_related","node_a_id":"n-9c0ed28aed848f75","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-9c0ed28aed848f75","other":{"id":"n-9c0ed28aed848f75","name":"basis_map_A013610 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["24","72","54"]],"card_rows":[["1"],["4","6"],["24","72","54"],["192","864","1296","648"]],"row_sums_preview":["1","10","150","3000","75000","2250000","78750000","3150000000"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013610"],"edges":["e-1eb90284df2d6b2d"],"text":"A156992 \u2192 basis A013610"},"headline":"A156992 \u2192 basis A013610","chain":null}},{"id":"l-1418247710ddd04f","kind":"fingerprint_related","node_a_id":"n-2b21a809f4a82963","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-2b21a809f4a82963","other":{"id":"n-2b21a809f4a82963","name":"reversal of basis_map_A013610 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["6","4"],["18","24","8"]],"card_rows":[["2"],["6","4"],["18","24","8"],["54","108","72","16"]],"row_sums_preview":["2","10","50","250","1250","6250","31250","156250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A007318","basis A013609"],"edges":["e-014dd8a0f9594ad5","e-49936077a24e6531"],"text":"A028326 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A028326 \u2192 basis A007318 \u2192 basis A013609","chain":null}},{"id":"l-151edf00a7a53e2b","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-bc910c5f6dde261c","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-bc910c5f6dde261c","other":{"id":"n-bc910c5f6dde261c","name":"basis_map_A013609 of basis_map_A013610 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","12"],["64","192","144"]],"card_rows":[["1"],["8","12"],["64","192","144"],["512","2304","3456","1728"]],"row_sums_preview":["1","20","400","8000","160000","3200000","64000000","1280000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["basis A013611","basis A013610"],"edges":["e-b3c62654af8090af","e-4bba97732cf4d306"],"text":"A027465 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A027465 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-156899a05dfb5532","kind":"fingerprint_related","node_a_id":"n-58c564fed7b3602a","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-58c564fed7b3602a","other":{"id":"n-58c564fed7b3602a","name":"reversal of basis_map_A007318 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["6","9"],["12","36","27"]],"card_rows":[["3"],["6","9"],["12","36","27"],["24","108","162","81"]],"row_sums_preview":["3","15","75","375","1875","9375","46875","234375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013609","reverse"],"edges":["e-618e7f7547136f14","e-af7cbf4a21d9b3ab"],"text":"A120910 \u2192 basis A013609 \u2192 reverse"},"headline":"A120910 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-16634cf04579da82","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-06ba959fdabb3333","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-06ba959fdabb3333","other":{"id":"n-06ba959fdabb3333","name":"basis_map_A013610 of shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["4","12","9"]],"card_rows":[["1"],["2","3"],["4","12","9"],["8","36","54","27"]],"row_sums_preview":["1","5","25","125","625","3125","15625","78125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A007318","basis A013610"],"edges":["e-279353e75dd63f59","e-198a418c186791ce"],"text":"basis-map coefficients of A026729 \u2192 basis A007318 \u2192 basis A013610"},"headline":"basis-map coefficients of A026729 \u2192 basis A007318 \u2192 basis A013610","chain":null}},{"id":"l-17fe5dd3137bb091","kind":"fingerprint_related","node_a_id":"n-9a254e2cf0e68e54","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-9a254e2cf0e68e54","other":{"id":"n-9a254e2cf0e68e54","name":"derivative of basis_map_A007318 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["36","24"]],"card_rows":[["6"],["36","24"],["162","216","72"],["648","1296","864","192"]],"row_sums_preview":["0","6","60","450","3000","18750","112500","656250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013609","D"],"edges":["e-618e7f7547136f14","e-9ffb865f688fca7a"],"text":"A120910 \u2192 basis A013609 \u2192 D"},"headline":"A120910 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-198a33ef4da59488","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-ba266de4de1b96ae","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-ba266de4de1b96ae","other":{"id":"n-ba266de4de1b96ae","name":"basis_map_A013610 of shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["4","12","9"],["8","36","54","27"],["16","96","216","216","81"]],"card_rows":[["4","12","9"],["8","36","54","27"],["16","96","216","216","81"],["32","240","720","1080","810","243"]],"row_sums_preview":["25","125","625","3125","15625","78125","390625","1953125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["x \u21a6 x+1","basis A013610"],"edges":["e-cb9b2e4b7b317e85","e-6f4385b1459f7e33"],"text":"basis-map coefficients of A112227 \u2192 x \u21a6 x+1 \u2192 basis A013610"},"headline":"basis-map coefficients of A112227 \u2192 x \u21a6 x+1 \u2192 basis A013610","chain":null}},{"id":"l-20db146c270485f5","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-fc842da63eb6b354","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-fc842da63eb6b354","other":{"id":"n-fc842da63eb6b354","name":"basis_map_A013610 of A Jacobsthal-Pascal triangle.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["12","36","27"]],"card_rows":[["1"],["2","3"],["12","36","27"],["40","180","270","135"]],"row_sums_preview":["1","5","75","625","6875","65625","671875","6640625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9f2158ef8512cadd","steps":["basis A013610"],"edges":["e-c429a2d88d818bb2"],"text":"A124860 \u2192 basis A013610"},"headline":"A124860 \u2192 basis A013610","chain":null}},{"id":"l-2340b39a1c43cbff","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-5b0a5a11faf67ad9","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-5b0a5a11faf67ad9","other":{"id":"n-5b0a5a11faf67ad9","name":"basis_map_A013610 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["324","432","144"],["5832","11664","7776","1728"],["104976","279936","279936","124416","20736"]],"card_rows":[["324","432","144"],["5832","11664","7776","1728"],["104976","279936","279936","124416","20736"],["1889568","6298560","8398080","5598720","1866240","248832"]],"row_sums_preview":["900","27000","810000","24300000","729000000","21870000000","656100000000","19683000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013609","basis A013610"],"edges":["e-ccf7b0982517b62a","e-18e95d52519ec86e"],"text":"A038328 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038328 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-262f899bb741a10c","kind":"fingerprint_related","node_a_id":"n-48d0de0097720f76","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-48d0de0097720f76","other":{"id":"n-48d0de0097720f76","name":"basis_map_A013610 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-3"],["4","12","9"]],"card_rows":[["1"],["-2","-3"],["4","12","9"],["-8","-36","-54","-27"]],"row_sums_preview":["1","-5","25","-125","625","-3125","15625","-78125"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A013610"],"edges":["e-e876d74ed42231b4"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A013610"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A013610","chain":null}},{"id":"l-29a4098cc9ce88b3","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-046fa60d0b9012ed","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-046fa60d0b9012ed","other":{"id":"n-046fa60d0b9012ed","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["9"],["108","162"]],"card_rows":[["9"],["108","162"],["972","2916","2187"],["7776","34992","52488","26244"]],"row_sums_preview":["0","9","270","6075","121500","2278125","41006250","717609375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013610","D"],"edges":["e-b88e4ed4246b1dea","e-7303854ae3cf4aba"],"text":"A038221 \u2192 basis A013610 \u2192 D"},"headline":"A038221 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-2a612eb59d4b353d","kind":"fingerprint_related","node_a_id":"n-48d2b46df5963045","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-48d2b46df5963045","other":{"id":"n-48d2b46df5963045","name":"basis_map_A013610 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","33"],["484","1452","1089"]],"card_rows":[["1"],["22","33"],["484","1452","1089"],["10648","47916","71874","35937"]],"row_sums_preview":["1","55","3025","166375","9150625","503284375","27680640625","1522435234375"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A013610"],"edges":["e-bb91b382acf43386"],"text":"A038325 \u2192 basis A013610"},"headline":"A038325 \u2192 basis A013610","chain":null}},{"id":"l-2d347b1f8d109774","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-5d17ea896d70e4da","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-5d17ea896d70e4da","other":{"id":"n-5d17ea896d70e4da","name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["24","16"],["360","480","160"]],"card_rows":[["1"],["24","16"],["360","480","160"],["4320","8640","5760","1280"]],"row_sums_preview":["1","40","1000","20000","350000","5600000","84000000","1200000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013611","basis A007318"],"edges":["e-159425c642d412bb","e-acbcb57301a85aa1"],"text":"A178820 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A178820 \u2192 basis A013611 \u2192 basis A007318","chain":null}},{"id":"l-2e92e2c284c9397b","kind":"fingerprint_related","node_a_id":"n-01cea33b6485d9cc","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-01cea33b6485d9cc","other":{"id":"n-01cea33b6485d9cc","name":"Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["9","12","4"]],"card_rows":[["1"],["3","2"],["9","12","4"],["27","54","36","8"]],"row_sums_preview":["1","5","25","125","625","3125","15625","78125"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038220"],"derivation":{"root":"n-01cea33b6485d9cc","steps":[],"edges":[],"text":"A038220"},"headline":"A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","chain":null}},{"id":"l-31cd36c83243db51","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-206b227355e1bd60","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-206b227355e1bd60","other":{"id":"n-206b227355e1bd60","name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","8"],["216","288","96"]],"card_rows":[["1"],["12","8"],["216","288","96"],["4320","8640","5760","1280"]],"row_sums_preview":["1","20","600","20000","700000","25200000","924000000","34320000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013611","basis A007318"],"edges":["e-6d76bdb8087cd7e0","e-42d66ad242812525"],"text":"A356546 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A356546 \u2192 basis A013611 \u2192 basis A007318","chain":null}},{"id":"l-352724d2f817dca3","kind":"fingerprint_related","node_a_id":"n-4f34a2344b135a41","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-4f34a2344b135a41","other":{"id":"n-4f34a2344b135a41","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["16","48","36"]],"card_rows":[["1"],["4","6"],["16","48","36"],["64","288","432","216"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A013610"],"edges":["e-1a1a6d136d357a71"],"text":"A038208 \u2192 basis A013610"},"headline":"A038208 \u2192 basis A013610","chain":null}},{"id":"l-387d5544fade836f","kind":"fingerprint_related","node_a_id":"n-1163fd2bfddb725c","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-1163fd2bfddb725c","other":{"id":"n-1163fd2bfddb725c","name":"shift_x_plus_one of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["27","18"],["243","324","108"]],"card_rows":[["3"],["27","18"],["243","324","108"],["2187","4374","2916","648"]],"row_sums_preview":["3","45","675","10125","151875","2278125","34171875","512578125"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-284e6c48cfe1aa93","e-7c224a86ed05e6d8"],"text":"A120909 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A120909 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-39591e7d813ee1a8","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-5733d79f45a14084","other":{"id":"n-5733d79f45a14084","name":"Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["4","12","9"]],"card_rows":[["1"],["2","3"],["4","12","9"],["8","36","54","27"]],"row_sums_preview":["1","5","25","125","625","3125","15625","78125"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A013620"],"derivation":{"root":"n-5733d79f45a14084","steps":[],"edges":[],"text":"A013620"},"headline":"A013620: Triangle of coefficients in expansion of (2+3x)^n.","chain":null}},{"id":"l-3963e07c3d5cc45c","kind":"fingerprint_related","node_a_id":"n-35f684e6e797bb02","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-35f684e6e797bb02","other":{"id":"n-35f684e6e797bb02","name":"basis_map_A013610 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["12","36","27"]],"card_rows":[["1"],["2","3"],["12","36","27"],["120","540","810","405"]],"row_sums_preview":["1","5","75","1875","65625","2953125","162421875","10557421875"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013610"],"edges":["e-0adda7d4e180ead0"],"text":"A085881 \u2192 basis A013610"},"headline":"A085881 \u2192 basis A013610","chain":null}},{"id":"l-3b08d0449012ac07","kind":"fingerprint_related","node_a_id":"n-98b538591e795174","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-98b538591e795174","other":{"id":"n-98b538591e795174","name":"shift_x_plus_one of basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","12","4"],["27","54","36","8"],["81","216","216","96","16"]],"card_rows":[["9","12","4"],["27","54","36","8"],["81","216","216","96","16"],["243","810","1080","720","240","32"]],"row_sums_preview":["25","125","625","3125","15625","78125","390625","1953125"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-7bee75fc05394bb8","e-25d850d9e17e91bf"],"text":"basis-map coefficients of A112227 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A112227 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-3d3a1a9f595c3d5e","kind":"fingerprint_related","node_a_id":"n-10e8604b45f3a91d","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-10e8604b45f3a91d","other":{"id":"n-10e8604b45f3a91d","name":"mobius_x_over_x_plus_one of shift_x_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","9"],["24","72","54"]],"card_rows":[["1"],["6","9"],["24","72","54"],["80","360","540","270"]],"row_sums_preview":["1","15","150","1250","9375","65625","437500","2812500"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-34e3d9a08d46fb54","e-d6b598c2dc3569db"],"text":"A094305 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A094305 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-3e922d23cd658123","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-ae4b5c81684d82cb","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-ae4b5c81684d82cb","other":{"id":"n-ae4b5c81684d82cb","name":"basis_map_A013609 of basis_map_A013610 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["32","96","72"]],"card_rows":[["1"],["4","6"],["32","96","72"],["384","1728","2592","1296"]],"row_sums_preview":["1","10","200","6000","240000","12000000","720000000","50400000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013610","basis A013609"],"edges":["e-77bf2cb365051314","e-54a4e22435302385"],"text":"A136572 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A136572 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-3f0283f516891ee5","kind":"fingerprint_related","node_a_id":"n-083700d21d46c394","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-083700d21d46c394","other":{"id":"n-083700d21d46c394","name":"shift_x_plus_one of basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-3","-2"],["9","12","4"]],"card_rows":[["1"],["-3","-2"],["9","12","4"],["-27","-54","-36","-8"]],"row_sums_preview":["1","-5","25","-125","625","-3125","15625","-78125"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-13dcb978bba71940","e-7063d79f350bc92a"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-409836d8400a2b61","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-17feb40af89fa354","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-17feb40af89fa354","other":{"id":"n-17feb40af89fa354","name":"basis_map_A013610 of derivative of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["4","6"]],"card_rows":[["-1"],["4","6"],["-12","-36","-27"],["32","144","216","108"]],"row_sums_preview":["0","-1","10","-75","500","-3125","18750","-109375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["D","basis A013610"],"edges":["e-3c99f80a26484c6d","e-9519140deeae3553"],"text":"basis-map coefficients of A113214, shifted \u2192 D \u2192 basis A013610"},"headline":"basis-map coefficients of A113214, shifted \u2192 D \u2192 basis A013610","chain":null}},{"id":"l-430fe35bcabf26f0","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-ea2a880b00e700bd","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-ea2a880b00e700bd","other":{"id":"n-ea2a880b00e700bd","name":"basis_map_A013610 of shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-3"],["4","12","9"]],"card_rows":[["1"],["-2","-3"],["4","12","9"],["-8","-36","-54","-27"]],"row_sums_preview":["1","-5","25","-125","625","-3125","15625","-78125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A007318","basis A013610"],"edges":["e-4c344817bd191837","e-f06931b803ccffb5"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A007318 \u2192 basis A013610"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A007318 \u2192 basis A013610","chain":null}},{"id":"l-48c24be24e04860f","kind":"fingerprint_related","node_a_id":"n-d5a5818819c0d79c","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-d5a5818819c0d79c","other":{"id":"n-d5a5818819c0d79c","name":"derivative of basis_map_A013610 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["24","36"]],"card_rows":[["6"],["24","36"],["72","216","162"],["192","864","1296","648"]],"row_sums_preview":["0","6","60","450","3000","18750","112500","656250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013610","D"],"edges":["e-28b777b3c0b25332","e-b25b6534deb2c937"],"text":"A028326 \u2192 basis A013610 \u2192 D"},"headline":"A028326 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-4d66e132116d06b4","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-edc6f5add0304545","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-edc6f5add0304545","other":{"id":"n-edc6f5add0304545","name":"basis_map_A013610 of T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["12","36","27"]],"card_rows":[["1"],["2","3"],["12","36","27"],["56","252","378","189"]],"row_sums_preview":["1","5","75","875","11875","159375","2203125","30703125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-63b9a6ee72661305","steps":["basis A013610"],"edges":["e-373d4e228900383c"],"text":"A376827 \u2192 basis A013610"},"headline":"A376827 \u2192 basis A013610","chain":null}},{"id":"l-4ff8f87d971f1b83","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-c8c02a416c4549d1","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-c8c02a416c4549d1","other":{"id":"n-c8c02a416c4549d1","name":"basis_map_A013609 of basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["33","22"],["1089","1452","484"]],"card_rows":[["1"],["33","22"],["1089","1452","484"],["35937","71874","47916","10648"]],"row_sums_preview":["1","55","3025","166375","9150625","503284375","27680640625","1522435234375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A007318","basis A013609"],"edges":["e-a5d0659f5315d3b7","e-1c85effd7da7d758"],"text":"A038325 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A038325 \u2192 basis A007318 \u2192 basis A013609","chain":null}},{"id":"l-52de0da220678f4e","kind":"fingerprint_related","node_a_id":"n-6874f381e9c7e0a2","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-6874f381e9c7e0a2","other":{"id":"n-6874f381e9c7e0a2","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","192","64"],["1728","3456","2304","512"],["20736","55296","55296","24576","4096"]],"card_rows":[["144","192","64"],["1728","3456","2304","512"],["20736","55296","55296","24576","4096"],["248832","829440","1105920","737280","245760","32768"]],"row_sums_preview":["400","8000","160000","3200000","64000000","1280000000","25600000000","512000000000"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":[],"edges":[],"text":"A038334"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","chain":null}},{"id":"l-541619af7c09a838","kind":"fingerprint_related","node_a_id":"n-49350b617f7163d1","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-49350b617f7163d1","other":{"id":"n-49350b617f7163d1","name":"basis_map_A013610 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["24","72","54"]],"card_rows":[["1"],["4","6"],["24","72","54"],["160","720","1080","540"]],"row_sums_preview":["1","10","150","2500","43750","787500","14437500","268125000"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013610"],"edges":["e-367cc697b56efa4f"],"text":"A356546 \u2192 basis A013610"},"headline":"A356546 \u2192 basis A013610","chain":null}},{"id":"l-5ffabb1b8a7b6e1b","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-270cc7b15b778f3a","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-270cc7b15b778f3a","other":{"id":"n-270cc7b15b778f3a","name":"basis_map_A013609 of basis_map_A013610 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["16","48","36"]],"card_rows":[["1"],["4","6"],["16","48","36"],["64","288","432","216"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013610","basis A013609"],"edges":["e-41e3c7da1a12c920","e-414785e3bf60dfbf"],"text":"basis-map coefficients of A026729 \u2192 basis A013610 \u2192 basis A013609"},"headline":"basis-map coefficients of A026729 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-6503cdbfaadffa37","kind":"fingerprint_related","node_a_id":"n-450d83dfd8c67163","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-450d83dfd8c67163","other":{"id":"n-450d83dfd8c67163","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["48","32"]],"card_rows":[["4"],["48","32"],["432","576","192"],["3456","6912","4608","1024"]],"row_sums_preview":["0","4","80","1200","16000","200000","2400000","28000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["basis A013611","D"],"edges":["e-0e13ca82ccc1a33c","e-c36df9e264073522"],"text":"A038243 \u2192 basis A013611 \u2192 D"},"headline":"A038243 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-6771307b94223eb7","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-3ce4733084222518","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-3ce4733084222518","other":{"id":"n-3ce4733084222518","name":"basis_map_A013610 of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","12"],["36","108","81"]],"card_rows":[["1"],["8","12"],["36","108","81"],["128","576","864","432"]],"row_sums_preview":["1","20","225","2000","15625","112500","765625","5000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["basis A013610"],"edges":["e-5008d121973b1f69"],"text":"A135065 \u2192 basis A013610"},"headline":"A135065 \u2192 basis A013610","chain":null}},{"id":"l-697ab13fa02c9868","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-c199d54e67c0f665","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-c199d54e67c0f665","other":{"id":"n-c199d54e67c0f665","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","9"],["28","84","63"]],"card_rows":[["1"],["6","9"],["28","84","63"],["120","540","810","405"]],"row_sums_preview":["1","15","175","1875","19375","196875","1984375","19921875"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d96419786c3ee2a","steps":["basis A013610"],"edges":["e-3119ad9ee9222212"],"text":"A134346 \u2192 basis A013610"},"headline":"A134346 \u2192 basis A013610","chain":null}},{"id":"l-6c5bd5b1b50f5fdb","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-9b745660022bc455","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-9b745660022bc455","other":{"id":"n-9b745660022bc455","name":"basis_map_A013610 of basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","12"],["96","288","216"]],"card_rows":[["1"],["8","12"],["96","288","216"],["1536","6912","10368","5184"]],"row_sums_preview":["1","20","600","24000","1200000","72000000","5040000000","403200000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013609","basis A013610"],"edges":["e-b7289e9f518eca68","e-fadcda75d3813ae8"],"text":"A156992 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A156992 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-6f43a4617e858844","kind":"fingerprint_related","node_a_id":"n-545ecce3e4f3660f","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-545ecce3e4f3660f","other":{"id":"n-545ecce3e4f3660f","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["8","24","18"]],"card_rows":[["1"],["4","6"],["8","24","18"],["32","144","216","108"]],"row_sums_preview":["1","10","50","500","2500","25000","125000","1250000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["basis A013610"],"edges":["e-9d8a38d997eb16b9"],"text":"A135838 \u2192 basis A013610"},"headline":"A135838 \u2192 basis A013610","chain":null}},{"id":"l-6f57ed9cadcbc460","kind":"fingerprint_related","node_a_id":"n-5c2409b346bb76f2","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-5c2409b346bb76f2","other":{"id":"n-5c2409b346bb76f2","name":"shift_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","6"],["81","108","36"]],"card_rows":[["1"],["9","6"],["81","108","36"],["729","1458","972","216"]],"row_sums_preview":["1","15","225","3375","50625","759375","11390625","170859375"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-14a3a792ab5fe5af","e-0723e3b257d7fb5b"],"text":"A013609 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A013609 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-6fc686295d69e71d","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-2a3d9797801799d0","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-2a3d9797801799d0","other":{"id":"n-2a3d9797801799d0","name":"reversal of basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","12"],["189","252","84"]],"card_rows":[["1"],["18","12"],["189","252","84"],["1512","3024","2016","448"]],"row_sums_preview":["1","30","525","7000","78750","787500","7218750","61875000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-a52727d7b51dd683","e-6843c78691aa8d1e"],"text":"A178822 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A178822 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-71f9a0e43faca90b","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-2c619e39fa0b3932","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-2c619e39fa0b3932","other":{"id":"n-2c619e39fa0b3932","name":"basis_map_A013610 of basis_map_A013609 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","12"],["96","288","216"]],"card_rows":[["1"],["8","12"],["96","288","216"],["1280","5760","8640","4320"]],"row_sums_preview":["1","20","600","20000","700000","25200000","924000000","34320000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013609","basis A013610"],"edges":["e-2e1c22b656cc26bb","e-6773f247ede0670b"],"text":"A356546 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A356546 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-75459bc5198d693e","kind":"fingerprint_related","node_a_id":"n-74c81a9d12bf4d45","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-74c81a9d12bf4d45","other":{"id":"n-74c81a9d12bf4d45","name":"derivative of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["12","18"]],"card_rows":[["3"],["12","18"],["36","108","81"],["96","432","648","324"]],"row_sums_preview":["0","3","30","225","1500","9375","56250","328125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["D"],"edges":["e-52ab624f3105e737"],"text":"A013620 \u2192 D"},"headline":"A013620 \u2192 D","chain":null}},{"id":"l-7915d3fe63f3e1f4","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-2c87f8e68aebecbb","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-2c87f8e68aebecbb","other":{"id":"n-2c87f8e68aebecbb","name":"basis_map_A013610 of Numbers in (4,4)-Pascal triangle .","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","12"],["16","48","36"]],"card_rows":[["1"],["8","12"],["16","48","36"],["32","144","216","108"]],"row_sums_preview":["1","20","100","500","2500","12500","62500","312500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3aa21f78720c1d0c","steps":["basis A013610"],"edges":["e-29017b2fb2c81ad5"],"text":"A132200 \u2192 basis A013610"},"headline":"A132200 \u2192 basis A013610","chain":null}},{"id":"l-7972bf15e5c0afc7","kind":"fingerprint_related","node_a_id":"n-5d099c0cb73e3d94","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-5d099c0cb73e3d94","other":{"id":"n-5d099c0cb73e3d94","name":"derivative of basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["12"],["120","180"]],"card_rows":[["12"],["120","180"],["720","2160","1620"],["3360","15120","22680","11340"]],"row_sums_preview":["0","12","300","4500","52500","525000","4725000","39375000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013610","D"],"edges":["e-24fdbe8219e7135e","e-1fa297788ff9c082"],"text":"A178820 \u2192 basis A013610 \u2192 D"},"headline":"A178820 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-79b47de3a0ead769","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-56b8c9249f5038fd","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-56b8c9249f5038fd","other":{"id":"n-56b8c9249f5038fd","name":"basis_map_A013610 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1024","3072","2304"],["32768","147456","221184","110592"],["1048576","6291456","14155776","14155776","5308416"]],"card_rows":[["1024","3072","2304"],["32768","147456","221184","110592"],["1048576","6291456","14155776","14155776","5308416"],["33554432","251658240","754974720","1132462080","849346560","254803968"]],"row_sums_preview":["6400","512000","40960000","3276800000","262144000000","20971520000000","1677721600000000","134217728000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013611","basis A013610"],"edges":["e-dab838a9194ad2c3","e-a20a24f0b8fdf1cd"],"text":"A038330 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A038330 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-7aeefb414b6f84ba","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-511785ca0b4480e5","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-511785ca0b4480e5","other":{"id":"n-511785ca0b4480e5","name":"reversal of basis_map_A013610 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","4"],["54","72","24"]],"card_rows":[["1"],["6","4"],["54","72","24"],["648","1296","864","192"]],"row_sums_preview":["1","10","150","3000","75000","2250000","78750000","3150000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A007318","basis A013609"],"edges":["e-bd56c468ffc76132","e-226d935e953bbdef"],"text":"A156992 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A156992 \u2192 basis A007318 \u2192 basis A013609","chain":null}},{"id":"l-7d503919e841c14a","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-0e1cd9b68048734e","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-0e1cd9b68048734e","other":{"id":"n-0e1cd9b68048734e","name":"derivative of basis_map_A013610 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["24","36"]],"card_rows":[["3"],["24","36"],["180","540","405"],["1344","6048","9072","4536"]],"row_sums_preview":["0","3","60","1125","21000","393750","7425000","140765625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013610","D"],"edges":["e-13e67b7b83b67a66","e-09f55821434e36b4"],"text":"A085880 \u2192 basis A013610 \u2192 D"},"headline":"A085880 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-7ec40218c65b7294","kind":"fingerprint_related","node_a_id":"n-0f5fb19aff15410b","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-0f5fb19aff15410b","other":{"id":"n-0f5fb19aff15410b","name":"basis_map_A013610 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["4","6"],["8","24","18"]],"card_rows":[["2"],["4","6"],["8","24","18"],["16","72","108","54"]],"row_sums_preview":["2","10","50","250","1250","6250","31250","156250"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013610"],"edges":["e-28b777b3c0b25332"],"text":"A028326 \u2192 basis A013610"},"headline":"A028326 \u2192 basis A013610","chain":null}},{"id":"l-7ff78a23ba01ccc8","kind":"fingerprint_related","node_a_id":"n-67e5bcee85c7fcd9","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-67e5bcee85c7fcd9","other":{"id":"n-67e5bcee85c7fcd9","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","1728","1296"],["13824","62208","93312","46656"],["331776","1990656","4478976","4478976","1679616"]],"card_rows":[["576","1728","1296"],["13824","62208","93312","46656"],["331776","1990656","4478976","4478976","1679616"],["7962624","59719680","179159040","268738560","201553920","60466176"]],"row_sums_preview":["3600","216000","12960000","777600000","46656000000","2799360000000","167961600000000","10077696000000000"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013610"],"edges":["e-85e8843cd1dca083"],"text":"A038338 \u2192 basis A013610"},"headline":"A038338 \u2192 basis A013610","chain":null}},{"id":"l-80ab6f2f42f21228","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-764cb1f9d7e8d694","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-764cb1f9d7e8d694","other":{"id":"n-764cb1f9d7e8d694","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["64","192","144"],["512","2304","3456","1728"],["4096","24576","55296","55296","20736"]],"card_rows":[["64","192","144"],["512","2304","3456","1728"],["4096","24576","55296","55296","20736"],["32768","245760","737280","1105920","829440","248832"]],"row_sums_preview":["400","8000","160000","3200000","64000000","1280000000","25600000000","512000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["reverse"],"edges":["e-9ea629408912a97a"],"text":"A038334 \u2192 reverse"},"headline":"A038334 \u2192 reverse","chain":null}},{"id":"l-8478fa3f8afd0863","kind":"fingerprint_related","node_a_id":"n-9dc2047d1a16a51a","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-9dc2047d1a16a51a","other":{"id":"n-9dc2047d1a16a51a","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","9"],["36","108","81"]],"card_rows":[["1"],["6","9"],["36","108","81"],["216","972","1458","729"]],"row_sums_preview":["1","15","225","3375","50625","759375","11390625","170859375"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013610"],"edges":["e-b88e4ed4246b1dea"],"text":"A038221 \u2192 basis A013610"},"headline":"A038221 \u2192 basis A013610","chain":null}},{"id":"l-85ede6da70de4436","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-889c040e1c837318","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-889c040e1c837318","other":{"id":"n-889c040e1c837318","name":"basis_map_A013609 of basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["27","36","12"]],"card_rows":[["1"],["3","2"],["27","36","12"],["405","810","540","120"]],"row_sums_preview":["1","5","75","1875","65625","2953125","162421875","10557421875"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A007318","basis A013609"],"edges":["e-5048ce3ebd740a6f","e-1bacdb4b39411576"],"text":"A085881 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A085881 \u2192 basis A007318 \u2192 basis A013609","chain":null}},{"id":"l-8a552f3c2cb9b7a3","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-53e421fb0a6e1da4","other":{"id":"n-53e421fb0a6e1da4","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","15"],["60","180","135"]],"card_rows":[["1"],["10","15"],["60","180","135"],["280","1260","1890","945"]],"row_sums_preview":["1","25","375","4375","43750","393750","3281250","25781250"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013610"],"edges":["e-0c010b1c5e20b0cb"],"text":"A178821 \u2192 basis A013610"},"headline":"A178821 \u2192 basis A013610","chain":null}},{"id":"l-8b325fb1ae9ab174","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-e5c39a3f8db2de85","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-e5c39a3f8db2de85","other":{"id":"n-e5c39a3f8db2de85","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","768","576"],["4096","18432","27648","13824"],["65536","393216","884736","884736","331776"]],"card_rows":[["256","768","576"],["4096","18432","27648","13824"],["65536","393216","884736","884736","331776"],["1048576","7864320","23592960","35389440","26542080","7962624"]],"row_sums_preview":["1600","64000","2560000","102400000","4096000000","163840000000","6553600000000","262144000000000"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-6efc8fa5932928ce","e-aedcf09c0d3938a8"],"text":"A038330 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038330 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-91d07dd95a81ec29","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-9992fd6674cf2f54","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-9992fd6674cf2f54","other":{"id":"n-9992fd6674cf2f54","name":"basis_map_A013610 of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["18","27"],["108","324","243"]],"card_rows":[["3"],["18","27"],["108","324","243"],["648","2916","4374","2187"]],"row_sums_preview":["3","45","675","10125","151875","2278125","34171875","512578125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013610","basis A013610"],"edges":["e-3ea97f7a94318269","e-72ee6ba6893bb2ad"],"text":"A120910 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A120910 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-91d56e588e81cb35","kind":"fingerprint_related","node_a_id":"n-95f27714049d8d62","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-95f27714049d8d62","other":{"id":"n-95f27714049d8d62","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","12"],["40","120","90"]],"card_rows":[["1"],["8","12"],["40","120","90"],["160","720","1080","540"]],"row_sums_preview":["1","20","250","2500","21875","175000","1312500","9375000"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013610"],"edges":["e-24fdbe8219e7135e"],"text":"A178820 \u2192 basis A013610"},"headline":"A178820 \u2192 basis A013610","chain":null}},{"id":"l-925ce82229420d46","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-66fb97bba5281255","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-66fb97bba5281255","other":{"id":"n-66fb97bba5281255","name":"shift_x_plus_one of basis_map_A013611 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","8"],["216","288","96"]],"card_rows":[["1"],["12","8"],["216","288","96"],["5184","10368","6912","1536"]],"row_sums_preview":["1","20","600","24000","1200000","72000000","5040000000","403200000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-413ab29272bdfecc","e-3d976880313de501"],"text":"A156992 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A156992 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-92a16a72827f16cf","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-bf590b1292365cf7","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-bf590b1292365cf7","other":{"id":"n-bf590b1292365cf7","name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["36","24"],["756","1008","336"]],"card_rows":[["1"],["36","24"],["756","1008","336"],["12096","24192","16128","3584"]],"row_sums_preview":["1","60","2100","56000","1260000","25200000","462000000","7920000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013611","basis A007318"],"edges":["e-377b2cf400ddb478","e-49baaa41f52ea25a"],"text":"A178822 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A178822 \u2192 basis A013611 \u2192 basis A007318","chain":null}},{"id":"l-a0ad9020f8eccb7f","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-5990edd3e1b831ed","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-5990edd3e1b831ed","other":{"id":"n-5990edd3e1b831ed","name":"basis_map_A013609 of basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","4"],["54","72","24"]],"card_rows":[["1"],["6","4"],["54","72","24"],["540","1080","720","160"]],"row_sums_preview":["1","10","150","2500","43750","787500","14437500","268125000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-0f8327ed3206c9f7","e-0b39650a16bfcd76"],"text":"A356546 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A356546 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-a15b3e1f902772ff","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-c79eb849b59e8079","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-c79eb849b59e8079","other":{"id":"n-c79eb849b59e8079","name":"basis_map_A013609 of basis_map_A013610 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["16","48","36"],["64","288","432","216"],["256","1536","3456","3456","1296"]],"card_rows":[["16","48","36"],["64","288","432","216"],["256","1536","3456","3456","1296"],["1024","7680","23040","34560","25920","7776"]],"row_sums_preview":["100","1000","10000","100000","1000000","10000000","100000000","1000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013610","basis A013609"],"edges":["e-33def10999a9e794","e-d79d30a1de7d614b"],"text":"basis-map coefficients of A112227 \u2192 basis A013610 \u2192 basis A013609"},"headline":"basis-map coefficients of A112227 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-a2ae9706b6a34d3f","kind":"fingerprint_related","node_a_id":"n-cdd67e6ac4684e40","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-cdd67e6ac4684e40","other":{"id":"n-cdd67e6ac4684e40","name":"basis_map_A013610 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["8","24","18"]],"card_rows":[["1"],["2","3"],["8","24","18"],["40","180","270","135"]],"row_sums_preview":["1","5","50","625","8750","131250","2062500","33515625"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013610"],"edges":["e-13e67b7b83b67a66"],"text":"A085880 \u2192 basis A013610"},"headline":"A085880 \u2192 basis A013610","chain":null}},{"id":"l-a40bcde681ecc9fa","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-59c0dcb7b3846aec","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-59c0dcb7b3846aec","other":{"id":"n-59c0dcb7b3846aec","name":"basis_map_A013609 of derivative of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["24","36"]],"card_rows":[["3"],["24","36"],["144","432","324"],["768","3456","5184","2592"]],"row_sums_preview":["0","3","60","900","12000","150000","1800000","21000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["D","basis A013609"],"edges":["e-5fa4095a1ef98485","e-1a96c89e451e8a04"],"text":"A013610 \u2192 D \u2192 basis A013609"},"headline":"A013610 \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-a6a931623aba4bdf","kind":"fingerprint_related","node_a_id":"n-5fded022f3086d45","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-5fded022f3086d45","other":{"id":"n-5fded022f3086d45","name":"derivative of basis_map_A013609 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["48","72"]],"card_rows":[["6"],["48","72"],["288","864","648"],["1536","6912","10368","5184"]],"row_sums_preview":["0","6","120","1800","24000","300000","3600000","42000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A013610","D"],"edges":["e-1a1a6d136d357a71","e-224f774559fef55e"],"text":"A038208 \u2192 basis A013610 \u2192 D"},"headline":"A038208 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-a7ad63b3bc33cfbf","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-f5bcaca8318904f4","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-f5bcaca8318904f4","other":{"id":"n-f5bcaca8318904f4","name":"basis_map_A013609 of basis_map_A013611 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","24"],["256","768","576"]],"card_rows":[["1"],["16","24"],["256","768","576"],["4096","18432","27648","13824"]],"row_sums_preview":["1","40","1600","64000","2560000","102400000","4096000000","163840000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["basis A013611","basis A013609"],"edges":["e-82b80af703a32f5e","e-c829f089025e6308"],"text":"A013610 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A013610 \u2192 basis A013611 \u2192 basis A013609","chain":null}},{"id":"l-a7b4f9c2bd591008","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-e06e654167609cff","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-e06e654167609cff","other":{"id":"n-e06e654167609cff","name":"basis_map_A013610 of shift_x_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","12"],["28","84","63"]],"card_rows":[["1"],["8","12"],["28","84","63"],["80","360","540","270"]],"row_sums_preview":["1","20","175","1250","8125","50000","296875","1718750"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["x \u21a6 x+1","basis A013610"],"edges":["e-28824f426538156a","e-f99cefb11baab9f9"],"text":"A131431 \u2192 x \u21a6 x+1 \u2192 basis A013610"},"headline":"A131431 \u2192 x \u21a6 x+1 \u2192 basis A013610","chain":null}},{"id":"l-a815f9cdf32cec0c","kind":"fingerprint_related","node_a_id":"n-c211a6e140527a22","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-c211a6e140527a22","other":{"id":"n-c211a6e140527a22","name":"basis_map_A013610 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["8","24","18"]],"card_rows":[["1"],["2","3"],["8","24","18"],["48","216","324","162"]],"row_sums_preview":["1","5","50","750","15000","375000","11250000","393750000"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013610"],"edges":["e-542183e09dec6678"],"text":"A196347 \u2192 basis A013610"},"headline":"A196347 \u2192 basis A013610","chain":null}},{"id":"l-aafe2118ecd523f9","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-6b935edaa11ea35d","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-6b935edaa11ea35d","other":{"id":"n-6b935edaa11ea35d","name":"basis_map_A013609 of derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["18","12"]],"card_rows":[["3"],["18","12"],["81","108","36"],["324","648","432","96"]],"row_sums_preview":["0","3","30","225","1500","9375","56250","328125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["D","basis A013609"],"edges":["e-d19cf3982d12ec2e","e-c9be2be264c3d4c7"],"text":"A120910 \u2192 D \u2192 basis A013609"},"headline":"A120910 \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-ab301e8baa44a376","kind":"fingerprint_related","node_a_id":"n-fd2004a7ef60a2e8","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-fd2004a7ef60a2e8","other":{"id":"n-fd2004a7ef60a2e8","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["12","8"]],"card_rows":[["2"],["12","8"],["54","72","24"],["216","432","288","64"]],"row_sums_preview":["0","2","20","150","1000","6250","37500","218750"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["D"],"edges":["e-ae35042602575497"],"text":"A038220 \u2192 D"},"headline":"A038220 \u2192 D","chain":null}},{"id":"l-ad492afa1f041895","kind":"fingerprint_related","node_a_id":"n-800c644c27747b67","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-800c644c27747b67","other":{"id":"n-800c644c27747b67","name":"shift_x_plus_one of basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-3","-2"],["9","12","4"]],"card_rows":[["1"],["-3","-2"],["9","12","4"],["-27","-54","-36","-8"]],"row_sums_preview":["1","-5","25","-125","625","-3125","15625","-78125"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013609","basis A007318"],"edges":["e-66f8525b021adc45","e-8fbc593398ab91d6"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013609 \u2192 basis A007318"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013609 \u2192 basis A007318","chain":null}},{"id":"l-b3a7d4b4a2a5f31b","kind":"fingerprint_related","node_a_id":"n-725ef1d8a9c0434d","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-725ef1d8a9c0434d","other":{"id":"n-725ef1d8a9c0434d","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["6","9"],["12","36","27"]],"card_rows":[["3"],["6","9"],["12","36","27"],["24","108","162","81"]],"row_sums_preview":["3","15","75","375","1875","9375","46875","234375"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["x \u21a6 x/(x+1)"],"edges":["e-8fbe952ff5bc6bbf"],"text":"A120910 \u2192 x \u21a6 x/(x+1)"},"headline":"A120910 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-b6a21b6f746a8ae5","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-56355adae11d1938","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-56355adae11d1938","other":{"id":"n-56355adae11d1938","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","12"],["64","192","144"]],"card_rows":[["1"],["8","12"],["64","192","144"],["512","2304","3456","1728"]],"row_sums_preview":["1","20","400","8000","160000","3200000","64000000","1280000000"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-267942f18014ef6e","e-4d5efa70f8c69814"],"text":"A027466 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A027466 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-baa5e056e481bf73","kind":"fingerprint_related","node_a_id":"n-9fce656f926a6332","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-9fce656f926a6332","other":{"id":"n-9fce656f926a6332","name":"mobius_x_over_x_plus_one of shift_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","12"],["36","108","81"]],"card_rows":[["1"],["8","12"],["36","108","81"],["128","576","864","432"]],"row_sums_preview":["1","20","225","2000","15625","112500","765625","5000000"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-18ace392e620d7bd","e-faeacd32ab14dcb6"],"text":"A135065 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A135065 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-bcde98f94491c569","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-4242efbbaf367697","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-4242efbbaf367697","other":{"id":"n-4242efbbaf367697","name":"basis_map_A013610 of basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","30"],["240","720","540"]],"card_rows":[["1"],["20","30"],["240","720","540"],["2240","10080","15120","7560"]],"row_sums_preview":["1","50","1500","35000","700000","12600000","210000000","3300000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013609","basis A013610"],"edges":["e-7c97406b114f1661","e-e3f9e03dd3f0e6a4"],"text":"A178821 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A178821 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-c1239baf3a5c5dab","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-1aa59385640eddd0","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-1aa59385640eddd0","other":{"id":"n-1aa59385640eddd0","name":"basis_map_A013610 of derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["8","12"]],"card_rows":[["2"],["8","12"],["24","72","54"],["64","288","432","216"]],"row_sums_preview":["0","2","20","150","1000","6250","37500","218750"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["D","basis A013610"],"edges":["e-509720341f277f4c","e-a9fc78c11d7d2a2c"],"text":"A028326 \u2192 D \u2192 basis A013610"},"headline":"A028326 \u2192 D \u2192 basis A013610","chain":null}},{"id":"l-c1394c72f4eca275","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-de18e467ab00c2f0","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-de18e467ab00c2f0","other":{"id":"n-de18e467ab00c2f0","name":"basis_map_A013610 of Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["8","24","18"]],"card_rows":[["1"],["4","6"],["8","24","18"],["16","72","108","54"]],"row_sums_preview":["1","10","50","250","1250","6250","31250","156250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f70486c726c7ad1c","steps":["basis A013610"],"edges":["e-03828cd5d01c1375"],"text":"A134058 \u2192 basis A013610"},"headline":"A134058 \u2192 basis A013610","chain":null}},{"id":"l-c16540d40f0eaf4e","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-2290e3a69a2374b9","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-2290e3a69a2374b9","other":{"id":"n-2290e3a69a2374b9","name":"basis_map_A013610 of basis_map_A013609 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["24","36"],["336","1008","756"]],"card_rows":[["1"],["24","36"],["336","1008","756"],["3584","16128","24192","12096"]],"row_sums_preview":["1","60","2100","56000","1260000","25200000","462000000","7920000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013609","basis A013610"],"edges":["e-5270021ea4ce6d61","e-65d47c9c10600bc7"],"text":"A178822 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A178822 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-c2cdebe2327a228a","kind":"fingerprint_related","node_a_id":"n-936114cbd4070c9a","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-936114cbd4070c9a","other":{"id":"n-936114cbd4070c9a","name":"basis_map_A013610 of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["30","45"],["180","540","405"]],"card_rows":[["3"],["30","45"],["180","540","405"],["840","3780","5670","2835"]],"row_sums_preview":["3","75","1125","13125","131250","1181250","9843750","77343750"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["basis A013610"],"edges":["e-127cb25e0d856e7f"],"text":"A344565 \u2192 basis A013610"},"headline":"A344565 \u2192 basis A013610","chain":null}},{"id":"l-c34052a47c966670","kind":"fingerprint_related","node_a_id":"n-000a80f5d182ab72","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-000a80f5d182ab72","other":{"id":"n-000a80f5d182ab72","name":"reversal of basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","8"],["90","120","40"]],"card_rows":[["1"],["12","8"],["90","120","40"],["540","1080","720","160"]],"row_sums_preview":["1","20","250","2500","21875","175000","1312500","9375000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A007318","basis A013609"],"edges":["e-2fa6de2302a1c958","e-6aa58465ff6b13a5"],"text":"A178820 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A178820 \u2192 basis A007318 \u2192 basis A013609","chain":null}},{"id":"l-c352cae682a5d069","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-3ab14e74c22de382","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-3ab14e74c22de382","other":{"id":"n-3ab14e74c22de382","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1728","2592"],["62208","186624","139968"],["1990656","8957952","13436928","6718464"]],"card_rows":[["1728","2592"],["62208","186624","139968"],["1990656","8957952","13436928","6718464"],["59719680","358318080","806215680","806215680","302330880"]],"row_sums_preview":["4320","388800","31104000","2332800000","167961600000","11757312000000","806215680000000","54419558400000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013610","D"],"edges":["e-85e8843cd1dca083","e-d1fb16f42a999d28"],"text":"A038338 \u2192 basis A013610 \u2192 D"},"headline":"A038338 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-c43342ba58f31e22","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-4c96f2a62fa07a0c","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-4c96f2a62fa07a0c","other":{"id":"n-4c96f2a62fa07a0c","name":"basis_map_A013610 of derivative of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["8","12"]],"card_rows":[["1"],["8","12"],["60","180","135"],["448","2016","3024","1512"]],"row_sums_preview":["0","1","20","375","7000","131250","2475000","46921875"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["D","basis A013610"],"edges":["e-592ee9fddf6f118c","e-81111ed60a6f8b0c"],"text":"A085880 \u2192 D \u2192 basis A013610"},"headline":"A085880 \u2192 D \u2192 basis A013610","chain":null}},{"id":"l-c4a8658ab5555aab","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-06e4f1e201575b5b","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-06e4f1e201575b5b","other":{"id":"n-06e4f1e201575b5b","name":"shift_x_plus_one of basis_map_A013611 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","4"],["72","96","32"]],"card_rows":[["1"],["6","4"],["72","96","32"],["1296","2592","1728","384"]],"row_sums_preview":["1","10","200","6000","240000","12000000","720000000","50400000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-b8ab5069d7db191f","e-aede6c73f57c33d3"],"text":"A196347 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A196347 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-c579d5e3cbd8486d","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-09536edb0cc841b5","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-09536edb0cc841b5","other":{"id":"n-09536edb0cc841b5","name":"basis_map_A013610 of Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","9"],["12","36","27"]],"card_rows":[["1"],["6","9"],["12","36","27"],["24","108","162","81"]],"row_sums_preview":["1","15","75","375","1875","9375","46875","234375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f71648c3ccd02d4","steps":["basis A013610"],"edges":["e-f2d079951b1f3d72"],"text":"A134059 \u2192 basis A013610"},"headline":"A134059 \u2192 basis A013610","chain":null}},{"id":"l-c841b28a05d7ce14","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-d8d5dbc1e1340fcb","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-d8d5dbc1e1340fcb","other":{"id":"n-d8d5dbc1e1340fcb","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1296","1728","576"],["46656","93312","62208","13824"],["1679616","4478976","4478976","1990656","331776"]],"card_rows":[["1296","1728","576"],["46656","93312","62208","13824"],["1679616","4478976","4478976","1990656","331776"],["60466176","201553920","268738560","179159040","59719680","7962624"]],"row_sums_preview":["3600","216000","12960000","777600000","46656000000","2799360000000","167961600000000","10077696000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013610","reverse"],"edges":["e-85e8843cd1dca083","e-a938ace1c829be95"],"text":"A038338 \u2192 basis A013610 \u2192 reverse"},"headline":"A038338 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-ca13016b86e7adb3","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-ac10b3f602dabf0c","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-ac10b3f602dabf0c","other":{"id":"n-ac10b3f602dabf0c","name":"shift_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","8"],["144","192","64"]],"card_rows":[["1"],["12","8"],["144","192","64"],["1728","3456","2304","512"]],"row_sums_preview":["1","20","400","8000","160000","3200000","64000000","1280000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","basis A013611"],"edges":["e-88ca34e9877f4f47","e-02e4478e9f124b98"],"text":"A038291 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A038291 \u2192 basis A013609 \u2192 basis A013611","chain":null}},{"id":"l-ca4aa7ed698383ab","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-f52aae032f81ced6","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-f52aae032f81ced6","other":{"id":"n-f52aae032f81ced6","name":"basis_map_A013610 of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","6"]],"card_rows":[["1"],["4","6"],["12","36","27"],["32","144","216","108"]],"row_sums_preview":["0","1","10","75","500","3125","18750","109375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-308cf60736924b4c","steps":["basis A013610"],"edges":["e-52e7f70645fda7c1"],"text":"A216973 \u2192 basis A013610"},"headline":"A216973 \u2192 basis A013610","chain":null}},{"id":"l-cb0e4ffa57ed2631","kind":"fingerprint_related","node_a_id":"n-60d5bc5c6c128acb","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-60d5bc5c6c128acb","other":{"id":"n-60d5bc5c6c128acb","name":"shift_x_plus_one of basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["9","12","4"]],"card_rows":[["1"],["3","2"],["9","12","4"],["27","54","36","8"]],"row_sums_preview":["1","5","25","125","625","3125","15625","78125"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-7869781294d5f3a8","e-0fb150afb461c14c"],"text":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-d1e9a76c0a976454","kind":"fingerprint_related","node_a_id":"n-408f7506d6594551","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-408f7506d6594551","other":{"id":"n-408f7506d6594551","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","4"],["36","48","16"]],"card_rows":[["1"],["6","4"],["36","48","16"],["216","432","288","64"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["basis A013611"],"edges":["e-0e13ca82ccc1a33c"],"text":"A038243 \u2192 basis A013611"},"headline":"A038243 \u2192 basis A013611","chain":null}},{"id":"l-d35c0882624bc2b0","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-1916a93c76107e9a","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-1916a93c76107e9a","other":{"id":"n-1916a93c76107e9a","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["18","24","8"]],"card_rows":[["1"],["3","2"],["18","24","8"],["162","324","216","48"]],"row_sums_preview":["1","5","50","750","15000","375000","11250000","393750000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-456dc355a6d78eff","e-1704930cf784c355"],"text":"A136572 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A136572 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-d3910705ffcf2b14","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-f5981aa8f519188f","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-f5981aa8f519188f","other":{"id":"n-f5981aa8f519188f","name":"shift_x_plus_one of basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","4"],["72","96","32"]],"card_rows":[["1"],["6","4"],["72","96","32"],["1080","2160","1440","320"]],"row_sums_preview":["1","10","200","5000","140000","4200000","132000000","4290000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-9f3d1c5f0f510549","e-07b4df9d18fdc788"],"text":"A085880 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A085880 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-d747864f5f0101ce","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-78e7224b9fa18f52","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-78e7224b9fa18f52","other":{"id":"n-78e7224b9fa18f52","name":"basis_map_A013610 of basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["8","12"],["32","96","72"]],"card_rows":[["2"],["8","12"],["32","96","72"],["128","576","864","432"]],"row_sums_preview":["2","20","200","2000","20000","200000","2000000","20000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013609","basis A013610"],"edges":["e-23f81812d1886d3d","e-3322a469340e0abb"],"text":"A028326 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A028326 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-da23bd200f41191c","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-c5577b9b62131d17","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-c5577b9b62131d17","other":{"id":"n-c5577b9b62131d17","name":"basis_map_A013609 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","768","256"],["13824","27648","18432","4096"],["331776","884736","884736","393216","65536"]],"card_rows":[["576","768","256"],["13824","27648","18432","4096"],["331776","884736","884736","393216","65536"],["7962624","26542080","35389440","23592960","7864320","1048576"]],"row_sums_preview":["1600","64000","2560000","102400000","4096000000","163840000000","6553600000000","262144000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013609","basis A013609"],"edges":["e-6efc8fa5932928ce","e-739c1c0c34c177e1"],"text":"A038330 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A038330 \u2192 basis A013609 \u2192 basis A013609","chain":null}},{"id":"l-dc2ce27a07c94227","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-a0b737d361865bc8","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-a0b737d361865bc8","other":{"id":"n-a0b737d361865bc8","name":"basis_map_A013610 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2304","6912","5184"],["110592","497664","746496","373248"],["5308416","31850496","71663616","71663616","26873856"]],"card_rows":[["2304","6912","5184"],["110592","497664","746496","373248"],["5308416","31850496","71663616","71663616","26873856"],["254803968","1911029760","5733089280","8599633920","6449725440","1934917632"]],"row_sums_preview":["14400","1728000","207360000","24883200000","2985984000000","358318080000000","42998169600000000","5159780352000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013609","basis A013610"],"edges":["e-9ec880266e74c89d","e-2727ac84351585b8"],"text":"A038338 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038338 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-dca26839fe731f7d","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-24b6838ef9a856ef","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-24b6838ef9a856ef","other":{"id":"n-24b6838ef9a856ef","name":"basis_map_A013610 of basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["8","24","18"]],"card_rows":[["1"],["2","3"],["8","24","18"],["24","108","162","81"]],"row_sums_preview":["1","5","50","375","3125","25000","203125","1640625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ed81f100961f7473","steps":["basis A013610"],"edges":["e-558494149d59eaec"],"text":"basis-map coefficients of A127647 \u2192 basis A013610"},"headline":"basis-map coefficients of A127647 \u2192 basis A013610","chain":null}},{"id":"l-dce231e432eac073","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-7b694ae3b14a3079","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-7b694ae3b14a3079","other":{"id":"n-7b694ae3b14a3079","name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["30","20"],["540","720","240"]],"card_rows":[["1"],["30","20"],["540","720","240"],["7560","15120","10080","2240"]],"row_sums_preview":["1","50","1500","35000","700000","12600000","210000000","3300000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013611","basis A007318"],"edges":["e-4faaf8fb320b6fca","e-761867a74fee9f28"],"text":"A178821 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A178821 \u2192 basis A013611 \u2192 basis A007318","chain":null}},{"id":"l-ddd5b2527182332e","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-f4c963fec6fd29eb","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-f4c963fec6fd29eb","other":{"id":"n-f4c963fec6fd29eb","name":"basis_map_A013610 of basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["44","66"],["1936","5808","4356"]],"card_rows":[["1"],["44","66"],["1936","5808","4356"],["85184","383328","574992","287496"]],"row_sums_preview":["1","110","12100","1331000","146410000","16105100000","1771561000000","194871710000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A013609","basis A013610"],"edges":["e-290484f08292dc0a","e-bc75fd1d87822836"],"text":"A038325 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038325 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-dfc146c2bbad323d","kind":"fingerprint_related","node_a_id":"n-8e6dfc463e261fa6","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-8e6dfc463e261fa6","other":{"id":"n-8e6dfc463e261fa6","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","18"],["84","252","189"]],"card_rows":[["1"],["12","18"],["84","252","189"],["448","2016","3024","1512"]],"row_sums_preview":["1","30","525","7000","78750","787500","7218750","61875000"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013610"],"edges":["e-3fa86ece9d0bedd1"],"text":"A178822 \u2192 basis A013610"},"headline":"A178822 \u2192 basis A013610","chain":null}},{"id":"l-e0b937f786341372","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-8649f595057a6925","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-8649f595057a6925","other":{"id":"n-8649f595057a6925","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["192","128"],["3456","4608","1536"],["55296","110592","73728","16384"]],"card_rows":[["192","128"],["3456","4608","1536"],["55296","110592","73728","16384"],["829440","2211840","2211840","983040","163840"]],"row_sums_preview":["320","9600","256000","6400000","153600000","3584000000","81920000000","1843200000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["D"],"edges":["e-25c972e7f2c368ab"],"text":"A038334 \u2192 D"},"headline":"A038334 \u2192 D","chain":null}},{"id":"l-e32d7f2402decda1","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-f58f0f323d989449","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-f58f0f323d989449","other":{"id":"n-f58f0f323d989449","name":"basis_map_A013610 of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["16","24"]],"card_rows":[["2"],["16","24"],["96","288","216"],["512","2304","3456","1728"]],"row_sums_preview":["0","2","40","600","8000","100000","1200000","14000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["D","basis A013610"],"edges":["e-a49f8ff31c907f66","e-44e188b545d43fd2"],"text":"A038208 \u2192 D \u2192 basis A013610"},"headline":"A038208 \u2192 D \u2192 basis A013610","chain":null}},{"id":"l-e702bd39be2edd81","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-1e792e6766e7cb36","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-1e792e6766e7cb36","other":{"id":"n-1e792e6766e7cb36","name":"basis_map_A013610 of basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["48","144","108"]],"card_rows":[["1"],["4","6"],["48","144","108"],["960","4320","6480","3240"]],"row_sums_preview":["1","10","300","15000","1050000","94500000","10395000000","1351350000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013609","basis A013610"],"edges":["e-0f3734a094503340","e-2edb99f8fc5447a9"],"text":"A085881 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A085881 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-e8a55fd82851f1a9","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-14200c6b33294c13","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-14200c6b33294c13","other":{"id":"n-14200c6b33294c13","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","9"],["24","72","54"]],"card_rows":[["1"],["6","9"],["24","72","54"],["80","360","540","270"]],"row_sums_preview":["1","15","150","1250","9375","65625","437500","2812500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["basis A013610"],"edges":["e-61eba3fc251318f8"],"text":"A094305 \u2192 basis A013610"},"headline":"A094305 \u2192 basis A013610","chain":null}},{"id":"l-eaa0bc018e91225c","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-09b0dafb5fde3569","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-09b0dafb5fde3569","other":{"id":"n-09b0dafb5fde3569","name":"derivative of basis_map_A013610 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["72","108"]],"card_rows":[["6"],["72","108"],["864","2592","1944"],["11520","51840","77760","38880"]],"row_sums_preview":["0","6","180","5400","180000","6750000","283500000","13230000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013610","D"],"edges":["e-1eb90284df2d6b2d","e-6be9bcfa4196c731"],"text":"A156992 \u2192 basis A013610 \u2192 D"},"headline":"A156992 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-eac62de1c3ee88b4","kind":"fingerprint_related","node_a_id":"n-653550d96f19f752","node_b_id":"n-f680a3b70a20ba67","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-653550d96f19f752","other":{"id":"n-653550d96f19f752","name":"shift_x_plus_one of basis_map_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["12","8"],["72","96","32"]],"card_rows":[["2"],["12","8"],["72","96","32"],["432","864","576","128"]],"row_sums_preview":["2","20","200","2000","20000","200000","2000000","20000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-3deac81ab612cd2e","e-6fb2690f326bbfdb"],"text":"A028326 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A028326 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-ec7a9f03b7a223d2","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-f1b84d684bc403ca","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-f1b84d684bc403ca","other":{"id":"n-f1b84d684bc403ca","name":"shift_x_plus_one of basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","4"],["108","144","48"]],"card_rows":[["1"],["6","4"],["108","144","48"],["3240","6480","4320","960"]],"row_sums_preview":["1","10","300","15000","1050000","94500000","10395000000","1351350000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013611","basis A007318"],"edges":["e-20dcbad557ba463b","e-25cffca751d997d9"],"text":"A085881 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A085881 \u2192 basis A013611 \u2192 basis A007318","chain":null}},{"id":"l-f0580a503a44889b","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-267a43bb721dbdb1","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-267a43bb721dbdb1","other":{"id":"n-267a43bb721dbdb1","name":"basis_map_A013610 of basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["32","96","72"]],"card_rows":[["1"],["4","6"],["32","96","72"],["320","1440","2160","1080"]],"row_sums_preview":["1","10","200","5000","140000","4200000","132000000","4290000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013609","basis A013610"],"edges":["e-ccc293aee69469d4","e-def8a3ad9bd5c098"],"text":"A085880 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A085880 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-f54b0e845b526f76","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-69e4e6e1680510e7","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-69e4e6e1680510e7","other":{"id":"n-69e4e6e1680510e7","name":"derivative of basis_map_A013610 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["24","36"]],"card_rows":[["3"],["24","36"],["216","648","486"],["2304","10368","15552","7776"]],"row_sums_preview":["0","3","60","1350","36000","1125000","40500000","1653750000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013610","D"],"edges":["e-542183e09dec6678","e-285f3014563a4f8d"],"text":"A196347 \u2192 basis A013610 \u2192 D"},"headline":"A196347 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-f5e9cf5c820530dc","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-eb695a28a5855a3c","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-eb695a28a5855a3c","other":{"id":"n-eb695a28a5855a3c","name":"reversal of basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","10"],["135","180","60"]],"card_rows":[["1"],["15","10"],["135","180","60"],["945","1890","1260","280"]],"row_sums_preview":["1","25","375","4375","43750","393750","3281250","25781250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013610","reverse"],"edges":["e-0c010b1c5e20b0cb","e-2b4dec5fe504b924"],"text":"A178821 \u2192 basis A013610 \u2192 reverse"},"headline":"A178821 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-f6ce4730c2104626","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-3769f49cc4c73730","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-3769f49cc4c73730","other":{"id":"n-3769f49cc4c73730","name":"basis_map_A013609 of basis_map_A013610 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-6"],["16","48","36"]],"card_rows":[["1"],["-4","-6"],["16","48","36"],["-64","-288","-432","-216"]],"row_sums_preview":["1","-10","100","-1000","10000","-100000","1000000","-10000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013610","basis A013609"],"edges":["e-fe4cc1f9dd1d3096","e-eff751bb2201d797"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013610 \u2192 basis A013609"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-f7636808b19375e5","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-b3568b7b03177313","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-b3568b7b03177313","other":{"id":"n-b3568b7b03177313","name":"shift_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","12"],["324","432","144"]],"card_rows":[["1"],["18","12"],["324","432","144"],["5832","11664","7776","1728"]],"row_sums_preview":["1","30","900","27000","810000","24300000","729000000","21870000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-fc7ebcb9132c171a","e-453a722df157c352"],"text":"A038221 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A038221 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-f89a44ec0e543a68","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-de683ecf1c59850f","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-de683ecf1c59850f","other":{"id":"n-de683ecf1c59850f","name":"basis_map_A013610 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","18"],["144","432","324"]],"card_rows":[["1"],["12","18"],["144","432","324"],["1728","7776","11664","5832"]],"row_sums_preview":["1","30","900","27000","810000","24300000","729000000","21870000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013609","basis A013610"],"edges":["e-86b1b19edf92ec9a","e-0346a66c4d830e16"],"text":"A038221 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038221 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-f9c670c76d9e8749","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-df799a0a0c7dce3f","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-df799a0a0c7dce3f","other":{"id":"n-df799a0a0c7dce3f","name":"basis_map_A013610 of derivative of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["12","18"],["48","144","108"],["160","720","1080","540"]],"row_sums_preview":["0","0","2","30","300","2500","18750","131250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["D","basis A013610"],"edges":["e-0d0bdaee7e5f71e7","e-fb5aaa325d3c173a"],"text":"A003506 \u2192 D \u2192 basis A013610"},"headline":"A003506 \u2192 D \u2192 basis A013610","chain":null}},{"id":"l-fa246b4a32f19aa1","kind":"fingerprint_related","node_a_id":"n-f680a3b70a20ba67","node_b_id":"n-e4771ce53e9351d5","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-e4771ce53e9351d5","other":{"id":"n-e4771ce53e9351d5","name":"basis_map_A013610 of Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","15"],["20","60","45"]],"card_rows":[["1"],["10","15"],["20","60","45"],["40","180","270","135"]],"row_sums_preview":["1","25","125","625","3125","15625","78125","390625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f0857c877d3df8d4","steps":["basis A013610"],"edges":["e-3c8af73c997dff75"],"text":"A135089 \u2192 basis A013610"},"headline":"A135089 \u2192 basis A013610","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:       0;\n  1:      15;\n  2:     180,      270;\n  3:    1260,     3780,      2835;\n  4:    6720,    30240,     45360,     22680;\n  5:   30240,   181440,    408240,    408240,     153090;\n  6:  120960,   907200,   2721600,   4082400,    3061800,     918540;\n  7:  443520,  3991680,  14968800,  29937600,   33679800,   20207880,    5051970;\n  8: 1520640, 15966720,  71850240, 179625600,  269438400,  242494560,  121247280,  25981560;\n  9: 4942080, 59304960, 311351040, 934053120, 1751349600, 2101619520, 1576214640, 675520560, 126660105;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n+8+(3*n+12)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 15.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 15;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 8 + (3*n + 12)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 15; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 8 + (3*n + 12)*x)*v[i - 1])/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013610","D"],"edges":["e-0c010b1c5e20b0cb","e-b66d4f3c62f1524f"],"text":"A178821 \u2192 basis A013610 \u2192 D"},"headline":"A178821 \u2192 basis A013610 \u2192 D","chain":null}