{"id":"n-7c4e320f16a7ccb1","revision":2,"first_index":0,"fingerprint":"f6-11aec07ce60b6d61","name":"shift_x_plus_one of Exponential Riordan array [x*exp(x),x].","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":[[],["1"],["4","2"],["12","12","3"],["32","48","24","4"],["80","160","120","40","5"],["192","480","480","240","60","6"],["448","1344","1680","1120","420","84","7"],["1024","3584","5376","4480","2240","672","112","8"],["2304","9216","16128","16128","10080","4032","1008","144","9"],["5120","23040","46080","53760","40320","20160","6720","1440","180","10"],["11264","56320","126720","168960","147840","88704","36960","10560","1980","220","11"],["24576","135168","337920","506880","506880","354816","177408","63360","15840","2640","264","12"],["53248","319488","878592","1464320","1647360","1317888","768768","329472","102960","22880","3432","312","13"],["114688","745472","2236416","4100096","5125120","4612608","3075072","1537536","576576","160160","32032","4368","364","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","6","27","108","405","1458","5103","17496","59049","196830","649539","2125764","6908733","22320522"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161380,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-d66a62e99e497f2a"]},{"id":161381,"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-d66a62e99e497f2a","e-fd845f603f8fe831"]},{"id":161382,"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-d66a62e99e497f2a","e-fd845f603f8fe831"]},{"id":161383,"position":3,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-1a916fd0c60fb95f"]},{"id":161384,"position":4,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-1a916fd0c60fb95f"]},{"id":161385,"position":5,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-1a916fd0c60fb95f"]},{"id":161386,"position":6,"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-fd845f603f8fe831"]},{"id":161387,"position":7,"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-fd845f603f8fe831"]}],"region_exact":{},"incoming_edges":[{"id":"e-1a916fd0c60fb95f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-7c4e320f16a7ccb1","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"f85e2b607bf7950e78f311bb3a3531ed9f7819f74f861e1cd0d64289be4e25bb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-7b5887f058255aa1","role":"sequence","name":"Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-d66a62e99e497f2a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-7c4e320f16a7ccb1","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"983b98274328e51fd8a6243fe3f9fa7c3f5f591ed82860b63c09268a3ad8f21c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-308cf60736924b4c","role":"sequence","name":"Exponential Riordan array [x*exp(x),x].","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]"}]},{"id":"e-fd845f603f8fe831","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-7c4e320f16a7ccb1","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":"b662a9094413b7cea3e79c0dfbd4a689760cfc9ac810da8ccb9484a4bd44de70","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-af754f07daf65b5e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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-009802e3a23f1262","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-66b456586fffaa5f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-66b456586fffaa5f","other":{"id":"n-66b456586fffaa5f","name":"shift_x_plus_one of Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["4","4","1"]],"card_rows":[["4","4","1"],["24","36","18","3"],["96","192","144","48","6"],["320","800","800","400","100","10"]],"row_sums_preview":["0","0","9","81","486","2430","10935","45927"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(\u2212(1/2)n+(1/2)n\u00b2)","x \u21a6 x+1"],"edges":["e-ccfad80d3bf960c1","e-7ca9df991456824b"],"text":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x+1"},"headline":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-0b53ea93587c5935","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-5d91f73259c0b712","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5d91f73259c0b712","other":{"id":"n-5d91f73259c0b712","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["40","40","10"]],"card_rows":[["1"],["8","4"],["40","40","10"],["160","240","120","20"]],"row_sums_preview":["1","12","90","540","2835","13608","61236","262440"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A007318"],"edges":["e-78029e34ed2e59f8"],"text":"A178820 \u2192 basis A007318"},"headline":"A178820 \u2192 basis A007318","chain":null}},{"id":"l-0e6ad74afeb88639","kind":"fingerprint_related","node_a_id":"n-416d26cbd480eee5","node_b_id":"n-7c4e320f16a7ccb1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-416d26cbd480eee5","other":{"id":"n-416d26cbd480eee5","name":"shift_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["4","2"],["8","8","2"]],"card_rows":[["2"],["4","2"],["8","8","2"],["16","24","12","2"]],"row_sums_preview":["2","6","18","54","162","486","1458","4374"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A007318"],"edges":["e-014dd8a0f9594ad5"],"text":"A028326 \u2192 basis A007318"},"headline":"A028326 \u2192 basis A007318","chain":null}},{"id":"l-13715d1023f67f94","kind":"fingerprint_related","node_a_id":"n-6ba906f5d673fec6","node_b_id":"n-7c4e320f16a7ccb1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-6ba906f5d673fec6","other":{"id":"n-6ba906f5d673fec6","name":"mobius_x_over_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["2","4"],["2","8","8"]],"card_rows":[["2"],["2","4"],["2","8","8"],["2","12","24","16"]],"row_sums_preview":["2","6","18","54","162","486","1458","4374"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["x \u21a6 x/(x+1)"],"edges":["e-71255c0d2ce9c4bc"],"text":"A028326 \u2192 x \u21a6 x/(x+1)"},"headline":"A028326 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-1796fda06559a9e1","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-9466c8441e24ce06","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-9466c8441e24ce06","other":{"id":"n-9466c8441e24ce06","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["1","4","4"]],"card_rows":[["1"],["-1","-2"],["1","4","4"],["-1","-6","-12","-8"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013609"],"edges":["e-13dcb978bba71940"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013609"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013609","chain":null}},{"id":"l-1e56915c3fa9c843","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-8f0d536701873284","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8f0d536701873284","other":{"id":"n-8f0d536701873284","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["8","32","32"]],"card_rows":[["1"],["4","8"],["8","32","32"],["32","192","384","256"]],"row_sums_preview":["1","12","72","864","5184","62208","373248","4478976"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["basis A013611"],"edges":["e-0f9b3c3168d8e647"],"text":"A135838 \u2192 basis A013611"},"headline":"A135838 \u2192 basis A013611","chain":null}},{"id":"l-1fb215aaf964af67","kind":"fingerprint_related","node_a_id":"n-af754f07daf65b5e","node_b_id":"n-7c4e320f16a7ccb1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-af754f07daf65b5e","other":{"id":"n-af754f07daf65b5e","name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["4","4","1"]],"card_rows":[["1"],["2","1"],["4","4","1"],["8","12","6","1"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038207"],"derivation":{"root":"n-af754f07daf65b5e","steps":[],"edges":[],"text":"A038207"},"headline":"A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","chain":null}},{"id":"l-1ffba4a562e20c71","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-076de49b14800916","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-076de49b14800916","other":{"id":"n-076de49b14800916","name":"derivative 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["12","24"]],"card_rows":[["6"],["12","24"],["18","72","72"],["24","144","288","192"]],"row_sums_preview":["0","6","36","162","648","2430","8748","30618"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["D"],"edges":["e-13c4b00cf881ee86"],"text":"A120909 \u2192 D"},"headline":"A120909 \u2192 D","chain":null}},{"id":"l-287ec0d37c171ecb","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-aa18ad1625904333","other":{"id":"n-aa18ad1625904333","name":"mobius_x_over_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["9","36","36"]],"card_rows":[["1"],["4","8"],["9","36","36"],["16","96","192","128"]],"row_sums_preview":["1","12","81","432","2025","8748","35721","139968"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(1+2n+n\u00b2)","x \u21a6 x/(x+1)"],"edges":["e-b78deffc64bc1942","e-963b3afafd9f7f70"],"text":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-2a7fc446342f7607","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-5317d1468ab69371","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5317d1468ab69371","other":{"id":"n-5317d1468ab69371","name":"basis_map_A013611 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["30","60"],["180","720","720"]],"card_rows":[["3"],["30","60"],["180","720","720"],["840","5040","10080","6720"]],"row_sums_preview":["3","90","1620","22680","272160","2939328","29393280","277136640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["basis A013611"],"edges":["e-8503bd9580ad59ab"],"text":"A344565 \u2192 basis A013611"},"headline":"A344565 \u2192 basis A013611","chain":null}},{"id":"l-2e0255905f192c77","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-0ea1c8d30da3abae","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0ea1c8d30da3abae","other":{"id":"n-0ea1c8d30da3abae","name":"mobius_x_over_x_plus_one of Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["1","4","4"]],"card_rows":[["1","4","4"],["3","18","36","24"],["6","48","144","192","96"],["10","100","400","800","800","320"]],"row_sums_preview":["0","0","9","81","486","2430","10935","45927"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(\u2212(1/2)n+(1/2)n\u00b2)","x \u21a6 x/(x+1)"],"edges":["e-ccfad80d3bf960c1","e-018032959df2ae1c"],"text":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-340d5cf1565c535b","kind":"fingerprint_related","node_a_id":"n-765f9e2b013b3c68","node_b_id":"n-7c4e320f16a7ccb1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-765f9e2b013b3c68","other":{"id":"n-765f9e2b013b3c68","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["3","6"],["3","12","12"]],"card_rows":[["3"],["3","6"],["3","12","12"],["3","18","36","24"]],"row_sums_preview":["3","9","27","81","243","729","2187","6561"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A120909"],"derivation":{"root":"n-765f9e2b013b3c68","steps":[],"edges":[],"text":"A120909"},"headline":"A120909: 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).","chain":null}},{"id":"l-3a5c59e6a94618bd","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-1e156f12069736a3","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-1e156f12069736a3","other":{"id":"n-1e156f12069736a3","name":"basis_map_A013611 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-4"],["4","16","16"]],"card_rows":[["1"],["-2","-4"],["4","16","16"],["-8","-48","-96","-64"]],"row_sums_preview":["1","-6","36","-216","1296","-7776","46656","-279936"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A013611"],"edges":["e-6ef42593efb6357c"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A013611"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A013611","chain":null}},{"id":"l-4351b2a324ec111e","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-8dc7b8a12ecdf1ca","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8dc7b8a12ecdf1ca","other":{"id":"n-8dc7b8a12ecdf1ca","name":"mobius_x_over_x_plus_one 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","8","8"]],"card_rows":[["1"],["1","2"],["2","8","8"],["5","30","60","40"]],"row_sums_preview":["1","3","18","135","1134","10206","96228","938223"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["x \u21a6 x/(x+1)"],"edges":["e-e1834b9d49e58efd"],"text":"A085880 \u2192 x \u21a6 x/(x+1)"},"headline":"A085880 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-4aec482b59221f30","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-8f9866a4e131e95d","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8f9866a4e131e95d","other":{"id":"n-8f9866a4e131e95d","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","11"],["484","484","121"]],"card_rows":[["1"],["22","11"],["484","484","121"],["10648","15972","7986","1331"]],"row_sums_preview":["1","33","1089","35937","1185921","39135393","1291467969","42618442977"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A007318"],"edges":["e-a5d0659f5315d3b7"],"text":"A038325 \u2192 basis A007318"},"headline":"A038325 \u2192 basis A007318","chain":null}},{"id":"l-4bd5a2371988d95f","kind":"fingerprint_related","node_a_id":"n-890099793b4c816b","node_b_id":"n-7c4e320f16a7ccb1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-890099793b4c816b","other":{"id":"n-890099793b4c816b","name":"mobius_x_over_x_plus_one of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","0","1"],["0","0","1","2"],["0","0","1","4","4"]],"card_rows":[["0","0","1"],["0","0","1","2"],["0","0","1","4","4"],["0","0","1","6","12","8"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["x \u21a6 x/(x+1)"],"edges":["e-dd3ac95d772228bd"],"text":"A159854 \u2192 x \u21a6 x/(x+1)"},"headline":"A159854 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-4ce2fed762745228","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-53ad300b8c449980","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-53ad300b8c449980","other":{"id":"n-53ad300b8c449980","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["9","36","36"]],"card_rows":[["1"],["3","6"],["9","36","36"],["27","162","324","216"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A013610"],"edges":["e-14a3a792ab5fe5af"],"text":"A013609 \u2192 basis A013610"},"headline":"A013609 \u2192 basis A013610","chain":null}},{"id":"l-4d9dc0af93285d9a","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-de403801a492a76b","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-de403801a492a76b","other":{"id":"n-de403801a492a76b","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","256","64"],["4096","6144","3072","512"],["65536","131072","98304","32768","4096"]],"card_rows":[["256","256","64"],["4096","6144","3072","512"],["65536","131072","98304","32768","4096"],["1048576","2621440","2621440","1310720","327680","32768"]],"row_sums_preview":["576","13824","331776","7962624","191102976","4586471424","110075314176","2641807540224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013609"],"edges":["e-6efc8fa5932928ce"],"text":"A038330 \u2192 basis A013609"},"headline":"A038330 \u2192 basis A013609","chain":null}},{"id":"l-50ce24dde6bc86e1","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-9805b07a338913cb","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-9805b07a338913cb","other":{"id":"n-9805b07a338913cb","name":"basis_map_A013611 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","44"],["484","1936","1936"]],"card_rows":[["1"],["22","44"],["484","1936","1936"],["10648","63888","127776","85184"]],"row_sums_preview":["1","66","4356","287496","18974736","1252332576","82653950016","5455160701056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A013611"],"edges":["e-6a2839a9935a683b"],"text":"A038325 \u2192 basis A013611"},"headline":"A038325 \u2192 basis A013611","chain":null}},{"id":"l-5c48c815cabd234a","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-5a242687ab527719","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5a242687ab527719","other":{"id":"n-5a242687ab527719","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","6"],["84","84","21"]],"card_rows":[["1"],["12","6"],["84","84","21"],["448","672","336","56"]],"row_sums_preview":["1","18","189","1512","10206","61236","336798","1732104"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["x \u21a6 x+1"],"edges":["e-a52727d7b51dd683"],"text":"A178822 \u2192 x \u21a6 x+1"},"headline":"A178822 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-5d007e7db5161a0a","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-feee915c2922c082","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-feee915c2922c082","other":{"id":"n-feee915c2922c082","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["36","144","144"]],"card_rows":[["1"],["6","12"],["36","144","144"],["216","1296","2592","1728"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013611"],"edges":["e-fc7ebcb9132c171a"],"text":"A038221 \u2192 basis A013611"},"headline":"A038221 \u2192 basis A013611","chain":null}},{"id":"l-5f210e0ea8cd8a54","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-ed2a02208588b66b","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ed2a02208588b66b","other":{"id":"n-ed2a02208588b66b","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["21","84","84"]],"card_rows":[["1"],["6","12"],["21","84","84"],["56","336","672","448"]],"row_sums_preview":["1","18","189","1512","10206","61236","336798","1732104"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["x \u21a6 x/(x+1)"],"edges":["e-376fcdbb8dac8b5b"],"text":"A178822 \u2192 x \u21a6 x/(x+1)"},"headline":"A178822 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-6133af26489d1a91","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-c80fa23f7424bb55","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-c80fa23f7424bb55","other":{"id":"n-c80fa23f7424bb55","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","4","4"]],"card_rows":[["1"],["1","2"],["1","4","4"],["1","6","12","8"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013609"],"edges":["e-7869781294d5f3a8"],"text":"basis-map coefficients of A026729 \u2192 basis A013609"},"headline":"basis-map coefficients of A026729 \u2192 basis A013609","chain":null}},{"id":"l-63a67404f18eecde","kind":"fingerprint_related","node_a_id":"n-71965e79e47a4526","node_b_id":"n-7c4e320f16a7ccb1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-71965e79e47a4526","other":{"id":"n-71965e79e47a4526","name":"Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","4","4"]],"card_rows":[["1"],["1","2"],["1","4","4"],["1","6","12","8"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A013609"],"derivation":{"root":"n-71965e79e47a4526","steps":[],"edges":[],"text":"A013609"},"headline":"A013609: Triangle of coefficients in expansion of (1+2*x)^n.","chain":null}},{"id":"l-640a3259e88dc714","kind":"fingerprint_related","node_a_id":"n-73bb59c333e132f0","node_b_id":"n-7c4e320f16a7ccb1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-73bb59c333e132f0","other":{"id":"n-73bb59c333e132f0","name":"mobius_x_over_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["1","4","4"]],"card_rows":[["1"],["-1","-2"],["1","4","4"],["-1","-6","-12","-8"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["x \u21a6 x/(x+1)"],"edges":["e-26d899ee6632b86c"],"text":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x/(x+1)"},"headline":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-658b2eeb9616ac58","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-63d7f3b1eb8cf1f2","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-63d7f3b1eb8cf1f2","other":{"id":"n-63d7f3b1eb8cf1f2","name":"shift_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["36","36","9"]],"card_rows":[["1"],["8","4"],["36","36","9"],["128","192","96","16"]],"row_sums_preview":["1","12","81","432","2025","8748","35721","139968"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(1+2n+n\u00b2)","x \u21a6 x+1"],"edges":["e-b78deffc64bc1942","e-18ace392e620d7bd"],"text":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 x \u21a6 x+1"},"headline":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-727d933a5faf0de8","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-986a623282b41276","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-986a623282b41276","other":{"id":"n-986a623282b41276","name":"basis_map_A013609 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["2","8","8"]],"card_rows":[["1"],["-1","-2"],["2","8","8"],["-6","-36","-72","-48"]],"row_sums_preview":["1","-3","18","-162","1944","-29160","524880","-11022480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["basis A013609"],"edges":["e-fbc81b67b4fab4f6"],"text":"basis-map coefficients of A021012 \u2192 basis A013609"},"headline":"basis-map coefficients of A021012 \u2192 basis A013609","chain":null}},{"id":"l-7bf9ab516be705e2","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-4dab44b954732fbb","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-4dab44b954732fbb","other":{"id":"n-4dab44b954732fbb","name":"basis_map_A013611 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["24","96","96"]],"card_rows":[["1"],["4","8"],["24","96","96"],["160","960","1920","1280"]],"row_sums_preview":["1","12","216","4320","90720","1959552","43110144","960740352"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013611"],"edges":["e-6d76bdb8087cd7e0"],"text":"A356546 \u2192 basis A013611"},"headline":"A356546 \u2192 basis A013611","chain":null}},{"id":"l-7c781522d79d14c5","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-a67bbce5c2713a1e","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a67bbce5c2713a1e","other":{"id":"n-a67bbce5c2713a1e","name":"basis_map_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["4","8"],["8","32","32"]],"card_rows":[["2"],["4","8"],["8","32","32"],["16","96","192","128"]],"row_sums_preview":["2","12","72","432","2592","15552","93312","559872"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013611"],"edges":["e-3deac81ab612cd2e"],"text":"A028326 \u2192 basis A013611"},"headline":"A028326 \u2192 basis A013611","chain":null}},{"id":"l-7fa7e9fda5ce5c8a","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-bc7333ad044cbf73","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-bc7333ad044cbf73","other":{"id":"n-bc7333ad044cbf73","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["64","64","16"]],"card_rows":[["1"],["8","4"],["64","64","16"],["512","768","384","64"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013611"],"edges":["e-267942f18014ef6e"],"text":"A027466 \u2192 basis A013611"},"headline":"A027466 \u2192 basis A013611","chain":null}},{"id":"l-8618a03aab817ebb","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-219653841046440e","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-219653841046440e","other":{"id":"n-219653841046440e","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","18"],["27","108","108"]],"card_rows":[["3"],["9","18"],["27","108","108"],["81","486","972","648"]],"row_sums_preview":["3","27","243","2187","19683","177147","1594323","14348907"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["basis A013610"],"edges":["e-284e6c48cfe1aa93"],"text":"A120909 \u2192 basis A013610"},"headline":"A120909 \u2192 basis A013610","chain":null}},{"id":"l-8b4f9ce47035ac8b","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-8d4637810467d09a","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8d4637810467d09a","other":{"id":"n-8d4637810467d09a","name":"basis_map_A007318 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["8","8","2"]],"card_rows":[["1"],["2","1"],["8","8","2"],["40","60","30","5"]],"row_sums_preview":["1","3","18","135","1134","10206","96228","938223"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["x \u21a6 x+1"],"edges":["e-48e55eb2c8b6f3ba"],"text":"A085880 \u2192 x \u21a6 x+1"},"headline":"A085880 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-8c741c3235333c01","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-3f5f8a84df712aee","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3f5f8a84df712aee","other":{"id":"n-3f5f8a84df712aee","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["8","8","2"]],"card_rows":[["1"],["4","2"],["8","8","2"],["32","48","24","4"]],"row_sums_preview":["1","6","18","108","324","1944","5832","34992"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["x \u21a6 x+1"],"edges":["e-7cade0dc9a919c2f"],"text":"A135838 \u2192 x \u21a6 x+1"},"headline":"A135838 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-8e9d94a0830374cc","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-2db2a1c44af41062","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-2db2a1c44af41062","other":{"id":"n-2db2a1c44af41062","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","576","144"],["13824","20736","10368","1728"],["331776","663552","497664","165888","20736"]],"card_rows":[["576","576","144"],["13824","20736","10368","1728"],["331776","663552","497664","165888","20736"],["7962624","19906560","19906560","9953280","2488320","248832"]],"row_sums_preview":["1296","46656","1679616","60466176","2176782336","78364164096","2821109907456","101559956668416"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A007318"],"edges":["e-a2ff444bbbdc7b83"],"text":"A038338 \u2192 basis A007318"},"headline":"A038338 \u2192 basis A007318","chain":null}},{"id":"l-92814b0013080bf3","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-439ac61e3f209ac0","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-439ac61e3f209ac0","other":{"id":"n-439ac61e3f209ac0","name":"basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","8","8"]],"card_rows":[["1"],["1","2"],["2","8","8"],["6","36","72","48"]],"row_sums_preview":["1","3","18","162","1944","29160","524880","11022480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013609"],"edges":["e-eea0c95a12d0633a"],"text":"A136572 \u2192 basis A013609"},"headline":"A136572 \u2192 basis A013609","chain":null}},{"id":"l-947708c2be91c919","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-a76db9b78dbb5551","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a76db9b78dbb5551","other":{"id":"n-a76db9b78dbb5551","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","24"],["84","336","336"]],"card_rows":[["1"],["12","24"],["84","336","336"],["448","2688","5376","3584"]],"row_sums_preview":["1","36","756","12096","163296","1959552","21555072","221709312"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013611"],"edges":["e-377b2cf400ddb478"],"text":"A178822 \u2192 basis A013611"},"headline":"A178822 \u2192 basis A013611","chain":null}},{"id":"l-9609fab12904321f","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-c201d29a62400440","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-c201d29a62400440","other":{"id":"n-c201d29a62400440","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","2304","2304"],["13824","82944","165888","110592"],["331776","2654208","7962624","10616832","5308416"]],"card_rows":[["576","2304","2304"],["13824","82944","165888","110592"],["331776","2654208","7962624","10616832","5308416"],["7962624","79626240","318504960","637009920","637009920","254803968"]],"row_sums_preview":["5184","373248","26873856","1934917632","139314069504","10030613004288","722204136308736","51998697814228992"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013611"],"edges":["e-c462762e7768ae01"],"text":"A038338 \u2192 basis A013611"},"headline":"A038338 \u2192 basis A013611","chain":null}},{"id":"l-9e58e7a196bdde33","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-8f1f80f146f45463","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8f1f80f146f45463","other":{"id":"n-8f1f80f146f45463","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["4","16","16"]],"card_rows":[["1"],["2","4"],["4","16","16"],["8","48","96","64"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["x \u21a6 x/(x+1)"],"edges":["e-19854cf600900932"],"text":"A038208 \u2192 x \u21a6 x/(x+1)"},"headline":"A038208 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-9eb9342468059c4e","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-af416c91e2aef37c","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-af416c91e2aef37c","other":{"id":"n-af416c91e2aef37c","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["24","24","6"]],"card_rows":[["1"],["6","3"],["24","24","6"],["80","120","60","10"]],"row_sums_preview":["1","9","54","270","1215","5103","20412","78732"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(1+(3/2)n+(1/2)n\u00b2)","x \u21a6 x+1"],"edges":["e-739fcf83152eb2a2","e-34e3d9a08d46fb54"],"text":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x+1"},"headline":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-a192f8f171efc95d","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-b061091a2d541bea","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-b061091a2d541bea","other":{"id":"n-b061091a2d541bea","name":"mobius_x_over_x_plus_one of Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["3","12","12"]],"card_rows":[["1"],["2","4"],["3","12","12"],["4","24","48","32"]],"row_sums_preview":["1","6","27","108","405","1458","5103","17496"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["D","x \u21a6 x/(x+1)"],"edges":["e-fd16b43297f2bb3e","e-bbc53a65557a79f5"],"text":"A007318 \u2192 D \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 D \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-a355936d57a9a453","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-fe7b03c8bdb96043","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fe7b03c8bdb96043","other":{"id":"n-fe7b03c8bdb96043","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","5"],["60","60","15"]],"card_rows":[["1"],["10","5"],["60","60","15"],["280","420","210","35"]],"row_sums_preview":["1","15","135","945","5670","30618","153090","721710"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A007318"],"edges":["e-484d449f4d0f99c5"],"text":"A178821 \u2192 basis A007318"},"headline":"A178821 \u2192 basis A007318","chain":null}},{"id":"l-a9596b9b1b24daee","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-465e4344a8d94925","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-465e4344a8d94925","other":{"id":"n-465e4344a8d94925","name":"basis_map_A013609 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["4","16","16"]],"card_rows":[["1"],["2","4"],["4","16","16"],["8","48","96","64"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013609"],"edges":["e-6abeab86de871564"],"text":"A130123 \u2192 basis A013609"},"headline":"A130123 \u2192 basis A013609","chain":null}},{"id":"l-b0d4cb82cd424dd7","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-3287a7cc800f9f98","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3287a7cc800f9f98","other":{"id":"n-3287a7cc800f9f98","name":"mobius_x_over_x_plus_one 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["15","30"],["45","180","180"]],"card_rows":[["3"],["15","30"],["45","180","180"],["105","630","1260","840"]],"row_sums_preview":["3","45","405","2835","17010","91854","459270","2165130"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["x \u21a6 x/(x+1)"],"edges":["e-3edca5aad3e7cb78"],"text":"A344565 \u2192 x \u21a6 x/(x+1)"},"headline":"A344565 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-b1e328982b607900","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-b4b87865f0ed6035","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-b4b87865f0ed6035","other":{"id":"n-b4b87865f0ed6035","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["12","48","48"]],"card_rows":[["1"],["2","4"],["12","48","48"],["120","720","1440","960"]],"row_sums_preview":["1","6","108","3240","136080","7348320","484989120","37829151360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013611"],"edges":["e-20dcbad557ba463b"],"text":"A085881 \u2192 basis A013611"},"headline":"A085881 \u2192 basis A013611","chain":null}},{"id":"l-b2167858170c1305","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-e8ae0bac7c0efceb","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e8ae0bac7c0efceb","other":{"id":"n-e8ae0bac7c0efceb","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["2","8","8"]],"card_rows":[["1"],["2","4"],["2","8","8"],["4","24","48","32"]],"row_sums_preview":["1","6","18","108","324","1944","5832","34992"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["x \u21a6 x/(x+1)"],"edges":["e-9f3b1e4aefe8044e"],"text":"A135838 \u2192 x \u21a6 x/(x+1)"},"headline":"A135838 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-b71cd0bff40ce779","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-e69b830457b341cf","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e69b830457b341cf","other":{"id":"n-e69b830457b341cf","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["8","32","32"]],"card_rows":[["1"],["2","4"],["8","32","32"],["40","240","480","320"]],"row_sums_preview":["1","6","72","1080","18144","326592","6158592","120092544"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013611"],"edges":["e-b498323f12f05355"],"text":"A085880 \u2192 basis A013611"},"headline":"A085880 \u2192 basis A013611","chain":null}},{"id":"l-b7e3b685fe9603b8","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-af521840f49b6d03","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-af521840f49b6d03","other":{"id":"n-af521840f49b6d03","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["12","6"]],"card_rows":[["3"],["12","6"],["36","36","9"],["96","144","72","12"]],"row_sums_preview":["0","3","18","81","324","1215","4374","15309"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["D"],"edges":["e-d19cf3982d12ec2e"],"text":"A120910 \u2192 D"},"headline":"A120910 \u2192 D","chain":null}},{"id":"l-bb71230e04f90308","kind":"fingerprint_related","node_a_id":"n-f8c9dbb903f44fdc","node_b_id":"n-7c4e320f16a7ccb1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-f8c9dbb903f44fdc","other":{"id":"n-f8c9dbb903f44fdc","name":"shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-1"],["4","4","1"]],"card_rows":[["1"],["-2","-1"],["4","4","1"],["-8","-12","-6","-1"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["x \u21a6 x+1"],"edges":["e-51a320c1f4e00092"],"text":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x+1","chain":null}},{"id":"l-c596872bd0e88c1d","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-0793a8fd2b196f0d","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0793a8fd2b196f0d","other":{"id":"n-0793a8fd2b196f0d","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["1","4","4"]],"card_rows":[["1"],["-1","-2"],["1","4","4"],["-1","-6","-12","-8"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013609"],"edges":["e-66f8525b021adc45"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013609"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013609","chain":null}},{"id":"l-c5c07d8cc13d6a36","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-3b6ec7300891f6df","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3b6ec7300891f6df","other":{"id":"n-3b6ec7300891f6df","name":"mobius_x_over_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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["6","24","24"]],"card_rows":[["1"],["3","6"],["6","24","24"],["10","60","120","80"]],"row_sums_preview":["1","9","54","270","1215","5103","20412","78732"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(1+(3/2)n+(1/2)n\u00b2)","x \u21a6 x/(x+1)"],"edges":["e-739fcf83152eb2a2","e-53cd3f44d77a4ed8"],"text":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-cabcf9d9999a607c","kind":"fingerprint_related","node_a_id":"n-51b4f5e7901c2b50","node_b_id":"n-7c4e320f16a7ccb1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-51b4f5e7901c2b50","other":{"id":"n-51b4f5e7901c2b50","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["6","3"],["12","12","3"]],"card_rows":[["3"],["6","3"],["12","12","3"],["24","36","18","3"]],"row_sums_preview":["3","9","27","81","243","729","2187","6561"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A120910"],"derivation":{"root":"n-51b4f5e7901c2b50","steps":[],"edges":[],"text":"A120910"},"headline":"A120910: 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.","chain":null}},{"id":"l-ccc8995630abe758","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-ec536fa823a7f7aa","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ec536fa823a7f7aa","other":{"id":"n-ec536fa823a7f7aa","name":"derivative of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["4","8"]],"card_rows":[["2"],["4","8"],["6","24","24"],["8","48","96","64"]],"row_sums_preview":["0","2","12","54","216","810","2916","10206"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["D"],"edges":["e-9f3f778aa1812acb"],"text":"A013609 \u2192 D"},"headline":"A013609 \u2192 D","chain":null}},{"id":"l-d0949ce670a175c5","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-df44f407d23e429d","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-df44f407d23e429d","other":{"id":"n-df44f407d23e429d","name":"basis_map_A007318 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["24","24","6"]],"card_rows":[["1"],["4","2"],["24","24","6"],["192","288","144","24"]],"row_sums_preview":["1","6","54","648","9720","174960","3674160","88179840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A007318"],"edges":["e-bd56c468ffc76132"],"text":"A156992 \u2192 basis A007318"},"headline":"A156992 \u2192 basis A007318","chain":null}},{"id":"l-d14488eee93ffbe1","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-77621b0cfa8249dd","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-77621b0cfa8249dd","other":{"id":"n-77621b0cfa8249dd","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["40","160","160"]],"card_rows":[["1"],["8","16"],["40","160","160"],["160","960","1920","1280"]],"row_sums_preview":["1","24","360","4320","45360","435456","3919104","33592320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013611"],"edges":["e-159425c642d412bb"],"text":"A178820 \u2192 basis A013611"},"headline":"A178820 \u2192 basis A013611","chain":null}},{"id":"l-d1ff5c111c942fc1","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-0b43db26e266f660","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0b43db26e266f660","other":{"id":"n-0b43db26e266f660","name":"basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["24","24","6"]],"card_rows":[["1"],["4","2"],["24","24","6"],["160","240","120","20"]],"row_sums_preview":["1","6","54","540","5670","61236","673596","7505784"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A007318"],"edges":["e-ced3e67c3d44f436"],"text":"A356546 \u2192 basis A007318"},"headline":"A356546 \u2192 basis A007318","chain":null}},{"id":"l-d202778870445dae","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-17166a4ad91393c8","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-17166a4ad91393c8","other":{"id":"n-17166a4ad91393c8","name":"basis_map_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["8","8","2"]],"card_rows":[["1"],["2","1"],["8","8","2"],["48","72","36","6"]],"row_sums_preview":["1","3","18","162","1944","29160","524880","11022480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A007318"],"edges":["e-703e907af1aa9666"],"text":"A196347 \u2192 basis A007318"},"headline":"A196347 \u2192 basis A007318","chain":null}},{"id":"l-d3d8f71557f0c9b7","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-0be19de8f4a58a03","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0be19de8f4a58a03","other":{"id":"n-0be19de8f4a58a03","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["16","64","64"]],"card_rows":[["1"],["4","8"],["16","64","64"],["64","384","768","512"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A013611"],"edges":["e-e6fcabb7b5383fc4"],"text":"A038208 \u2192 basis A013611"},"headline":"A038208 \u2192 basis A013611","chain":null}},{"id":"l-d3e4d208017f2de7","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-11dd292935cd7dbf","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-11dd292935cd7dbf","other":{"id":"n-11dd292935cd7dbf","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["10","40","40"]],"card_rows":[["1"],["4","8"],["10","40","40"],["20","120","240","160"]],"row_sums_preview":["1","12","90","540","2835","13608","61236","262440"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["x \u21a6 x/(x+1)"],"edges":["e-384d0b9c26a615c7"],"text":"A178820 \u2192 x \u21a6 x/(x+1)"},"headline":"A178820 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-d8a603a6334f22b0","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-7f05fd9e9c6948d8","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-7f05fd9e9c6948d8","other":{"id":"n-7f05fd9e9c6948d8","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["36","36","9"]],"card_rows":[["1"],["6","3"],["36","36","9"],["216","324","162","27"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["basis A013610"],"edges":["e-2d1052dd37eda11b"],"text":"A038243 \u2192 basis A013610"},"headline":"A038243 \u2192 basis A013610","chain":null}},{"id":"l-d9d77d55ae5c7189","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-03ac40ba4fd62e4c","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-03ac40ba4fd62e4c","other":{"id":"n-03ac40ba4fd62e4c","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["12","12","3"]],"card_rows":[["1"],["2","1"],["12","12","3"],["120","180","90","15"]],"row_sums_preview":["1","3","27","405","8505","229635","7577955","295540245"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A007318"],"edges":["e-5048ce3ebd740a6f"],"text":"A085881 \u2192 basis A007318"},"headline":"A085881 \u2192 basis A007318","chain":null}},{"id":"l-da426c50c7eca298","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-4766ad62f3b3a125","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-4766ad62f3b3a125","other":{"id":"n-4766ad62f3b3a125","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["24","96","96"]],"card_rows":[["1"],["4","8"],["24","96","96"],["192","1152","2304","1536"]],"row_sums_preview":["1","12","216","5184","155520","5598720","235146240","11287019520"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013611"],"edges":["e-fb2d67190f0fa658"],"text":"A156992 \u2192 basis A013611"},"headline":"A156992 \u2192 basis A013611","chain":null}},{"id":"l-db64a2e97ac3b91f","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-fff9dc1c6c2891b5","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fff9dc1c6c2891b5","other":{"id":"n-fff9dc1c6c2891b5","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["16","16","4"]],"card_rows":[["1"],["4","2"],["16","16","4"],["64","96","48","8"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A007318"],"edges":["e-1756f066a00732e3"],"text":"A038208 \u2192 basis A007318"},"headline":"A038208 \u2192 basis A007318","chain":null}},{"id":"l-e0b3a8d7e763f984","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-fbc103865e18ab22","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fbc103865e18ab22","other":{"id":"n-fbc103865e18ab22","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","10"],["15","60","60"]],"card_rows":[["1"],["5","10"],["15","60","60"],["35","210","420","280"]],"row_sums_preview":["1","15","135","945","5670","30618","153090","721710"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["x \u21a6 x/(x+1)"],"edges":["e-da74bc8fbb9fce7e"],"text":"A178821 \u2192 x \u21a6 x/(x+1)"},"headline":"A178821 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-e777f59d27d7e610","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-fdb3be84f583ed95","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fdb3be84f583ed95","other":{"id":"n-fdb3be84f583ed95","name":"basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","4","4"],["1","6","12","8"],["1","8","24","32","16"]],"card_rows":[["1","4","4"],["1","6","12","8"],["1","8","24","32","16"],["1","10","40","80","80","32"]],"row_sums_preview":["9","27","81","243","729","2187","6561","19683"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013609"],"edges":["e-7bee75fc05394bb8"],"text":"basis-map coefficients of A112227 \u2192 basis A013609"},"headline":"basis-map coefficients of A112227 \u2192 basis A013609","chain":null}},{"id":"l-f259d70f9133801a","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-ce701caf7dd0b8e7","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ce701caf7dd0b8e7","other":{"id":"n-ce701caf7dd0b8e7","name":"basis_map_A013611 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["8","32","32"]],"card_rows":[["1"],["2","4"],["8","32","32"],["48","288","576","384"]],"row_sums_preview":["1","6","72","1296","31104","933120","33592320","1410877440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013611"],"edges":["e-f21031c0731e2731"],"text":"A196347 \u2192 basis A013611"},"headline":"A196347 \u2192 basis A013611","chain":null}},{"id":"l-f68eef6d52c90177","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-e93f0e54dbc33ad5","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e93f0e54dbc33ad5","other":{"id":"n-e93f0e54dbc33ad5","name":"basis_map_A007318 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["30","15"],["180","180","45"]],"card_rows":[["3"],["30","15"],["180","180","45"],["840","1260","630","105"]],"row_sums_preview":["3","45","405","2835","17010","91854","459270","2165130"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["basis A007318"],"edges":["e-5d53734a383a65e0"],"text":"A344565 \u2192 basis A007318"},"headline":"A344565 \u2192 basis A007318","chain":null}},{"id":"l-f97343fd93f69129","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-57f251ac09effa9f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-57f251ac09effa9f","other":{"id":"n-57f251ac09effa9f","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","20"],["60","240","240"]],"card_rows":[["1"],["10","20"],["60","240","240"],["280","1680","3360","2240"]],"row_sums_preview":["1","30","540","7560","90720","979776","9797760","92378880"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013611"],"edges":["e-75b8144fcd4b7728"],"text":"A178821 \u2192 basis A013611"},"headline":"A178821 \u2192 basis A013611","chain":null}},{"id":"l-451ba7901a7bfa95","kind":"oeis_match","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":null,"fingerprint":null,"oeis_id":"A027471","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A027471","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A027471","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-7c4e320f16a7ccb1","oeis_id":"A027471","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":10764}],"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:    1;\n  2:    4,    2;\n  3:   12,   12,     3;\n  4:   32,   48,    24,     4;\n  5:   80,  160,   120,    40,     5;\n  6:  192,  480,   480,   240,    60,    6;\n  7:  448, 1344,  1680,  1120,   420,   84,    7;\n  8: 1024, 3584,  5376,  4480,  2240,  672,  112,   8;\n  9: 2304, 9216, 16128, 16128, 10080, 4032, 1008, 144, 9;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n+n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + n*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + n*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-af754f07daf65b5e","steps":["D"],"edges":["e-fd845f603f8fe831"],"text":"A038207 \u2192 D"},"headline":"A038207 \u2192 D","chain":null}