{"id":"n-408f7506d6594551","revision":1,"first_index":0,"fingerprint":"f6-f06e51c139cfae05","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["6","4"],["36","48","16"],["216","432","288","64"],["1296","3456","3456","1536","256"],["7776","25920","34560","23040","7680","1024"],["46656","186624","311040","276480","138240","36864","4096"],["279936","1306368","2612736","2903040","1935360","774144","172032","16384"],["1679616","8957952","20901888","27869184","23224320","12386304","4128768","786432","65536"],["10077696","60466176","161243136","250822656","250822656","167215104","74317824","21233664","3538944","262144"],["60466176","403107840","1209323520","2149908480","2508226560","2006581248","1114767360","424673280","106168320","15728640","1048576"],["362797056","2660511744","8868372480","17736744960","23648993280","22072393728","14714929152","7007109120","2335703040","519045120","69206016","4194304"],["2176782336","17414258688","63852281856","141893959680","212840939520","227030335488","176579149824","100902371328","42042654720","12457082880","2491416576","301989888","16777216"],["13060694016","113192681472","452770725888","1106772885504","1844621475840","2213545771008","1967596240896","1311730827264","655865413632","242913116160","64776830976","11777605632","1308622848","67108864"],["78364164096","731398864896","3169395081216","8451720216576","15494820397056","20659760529408","20659760529408","15740769927168","9182115790848","4080940351488","1360313450496","329772957696","54962159616","5637144576","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","100","1000","10000","100000","1000000","10000000","100000000","1000000000","10000000000","100000000000","1000000000000","10000000000000","100000000000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168080,"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-0e13ca82ccc1a33c"]},{"id":168081,"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-0e13ca82ccc1a33c"]},{"id":168082,"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-0e13ca82ccc1a33c"]}],"region_exact":{},"incoming_edges":[{"id":"e-0e13ca82ccc1a33c","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-408f7506d6594551","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"ae86af9c5af7be8b5ec9d16e7ddd47548ba8bba983a1e5a1c1fd858e8776d149","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7b91a7a037fb3513","role":"sequence","name":"Triangle whose (i,j)-th entry is 5^(i-j)*binomial(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-094873eccfc665fb","kind":"fingerprint_related","node_a_id":"n-67e5bcee85c7fcd9","node_b_id":"n-408f7506d6594551","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]"],"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-09cf74b3a0c36708","kind":"fingerprint_related","node_a_id":"n-4f34a2344b135a41","node_b_id":"n-408f7506d6594551","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]"],"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-1ef33c98b4dcbdb7","kind":"fingerprint_related","node_a_id":"n-545ecce3e4f3660f","node_b_id":"n-408f7506d6594551","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-2141a2992af2c0c8","kind":"fingerprint_related","node_a_id":"n-9dc2047d1a16a51a","node_b_id":"n-408f7506d6594551","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]"],"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-23c9f59bbfa8d7ef","kind":"fingerprint_related","node_a_id":"n-9c0ed28aed848f75","node_b_id":"n-408f7506d6594551","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]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013610"],"edges":["e-d66805981c716c2b"],"text":"A156992 \u2192 basis A013610"},"headline":"A156992 \u2192 basis A013610","chain":null}},{"id":"l-247d1e7cdecbe06d","kind":"fingerprint_related","node_a_id":"n-6874f381e9c7e0a2","node_b_id":"n-408f7506d6594551","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-3075a952c0ad90cf","kind":"fingerprint_related","node_a_id":"n-c211a6e140527a22","node_b_id":"n-408f7506d6594551","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]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013610"],"edges":["e-d8646d207d983420"],"text":"A196347 \u2192 basis A013610"},"headline":"A196347 \u2192 basis A013610","chain":null}},{"id":"l-37d4ad689f3acef6","kind":"fingerprint_related","node_a_id":"n-48d2b46df5963045","node_b_id":"n-408f7506d6594551","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]"],"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-409eff94dd38155e","kind":"fingerprint_related","node_a_id":"n-95f27714049d8d62","node_b_id":"n-408f7506d6594551","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]"],"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-4b0978f21318a7db","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-408f7506d6594551","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-5efd7f1cd5654208","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","node_b_id":"n-408f7506d6594551","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]"],"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-659c7641b87b74e4","kind":"fingerprint_related","node_a_id":"n-cdd67e6ac4684e40","node_b_id":"n-408f7506d6594551","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]"],"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-660b4bdb102c02df","kind":"fingerprint_related","node_a_id":"n-35f684e6e797bb02","node_b_id":"n-408f7506d6594551","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]"],"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-8a5efae2374f2cfb","kind":"fingerprint_related","node_a_id":"n-f6495b2c898daf02","node_b_id":"n-408f7506d6594551","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-8ca2d9c13c7d0325","kind":"fingerprint_related","node_a_id":"n-fd2004a7ef60a2e8","node_b_id":"n-408f7506d6594551","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]"],"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-97e510dc2e7303b0","kind":"fingerprint_related","node_a_id":"n-01cea33b6485d9cc","node_b_id":"n-408f7506d6594551","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-ae8f392084c245cf","kind":"fingerprint_related","node_a_id":"n-48d0de0097720f76","node_b_id":"n-408f7506d6594551","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]"],"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-bfbeed4fcf546cdd","kind":"fingerprint_related","node_a_id":"n-74c81a9d12bf4d45","node_b_id":"n-408f7506d6594551","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-e5d8095b121a37d0","kind":"fingerprint_related","node_a_id":"n-725ef1d8a9c0434d","node_b_id":"n-408f7506d6594551","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-e79f17bf084a062b","kind":"fingerprint_related","node_a_id":"n-49350b617f7163d1","node_b_id":"n-408f7506d6594551","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]"],"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-ed34c575bd9ccbf6","kind":"fingerprint_related","node_a_id":"n-8e6dfc463e261fa6","node_b_id":"n-408f7506d6594551","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]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013610"],"edges":["e-97dd7dd90f768ab3"],"text":"A178822 \u2192 basis A013610"},"headline":"A178822 \u2192 basis A013610","chain":null}},{"id":"l-f3cb7c7440594ac9","kind":"fingerprint_related","node_a_id":"n-936114cbd4070c9a","node_b_id":"n-408f7506d6594551","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-f73190eee6041707","kind":"fingerprint_related","node_a_id":"n-0f5fb19aff15410b","node_b_id":"n-408f7506d6594551","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]"],"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-105cbfa6c1f7ee4f","kind":"oeis_match","node_a_id":"n-408f7506d6594551","node_b_id":null,"fingerprint":null,"oeis_id":"A011557","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":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A011557","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-21ac57f7876e840f","kind":"oeis_match","node_a_id":"n-408f7506d6594551","node_b_id":null,"fingerprint":null,"oeis_id":"A155148","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A155148","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ae2c9b55b4ed85d9","kind":"oeis_match","node_a_id":"n-408f7506d6594551","node_b_id":null,"fingerprint":null,"oeis_id":"A038258","in_catalog":0,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":36},"in_catalog":false,"oeis_id":"A038258","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d27241874a312a84","kind":"oeis_match","node_a_id":"n-408f7506d6594551","node_b_id":null,"fingerprint":null,"oeis_id":"A178500","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A178500","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-dbb6958a9949fb96","kind":"oeis_match","node_a_id":"n-408f7506d6594551","node_b_id":null,"fingerprint":null,"oeis_id":"A038236","in_catalog":0,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"reversed_triangle","terms_compared":36},"in_catalog":false,"oeis_id":"A038236","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-eb98a92cb883bf2f","kind":"oeis_match","node_a_id":"n-408f7506d6594551","node_b_id":null,"fingerprint":null,"oeis_id":"A178501","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":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A178501","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A038236","relation":"match_reversed_triangle"},{"oeis_id":"A011557","relation":"match_row_sums"},{"oeis_id":"A155148","relation":"match_row_sums"},{"oeis_id":"A178500","relation":"match_row_sums"},{"oeis_id":"A178501","relation":"match_row_sums"},{"oeis_id":"A038258","relation":"match_triangle"}],"bfiles":[{"node_id":"n-408f7506d6594551","oeis_id":"A011557","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":19,"oeis_terms_available":19,"term_count":200,"raw_size":20990},{"node_id":"n-408f7506d6594551","oeis_id":"A038236","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":36,"oeis_terms_agreeing":36,"oeis_terms_available":36,"term_count":990,"raw_size":30244},{"node_id":"n-408f7506d6594551","oeis_id":"A038258","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":36,"oeis_terms_agreeing":36,"oeis_terms_available":36,"term_count":990,"raw_size":30244}],"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:        1;\n  1:        6,        4;\n  2:       36,       48,        16;\n  3:      216,      432,       288,        64;\n  4:     1296,     3456,      3456,      1536,       256;\n  5:     7776,    25920,     34560,     23040,      7680,      1024;\n  6:    46656,   186624,    311040,    276480,    138240,     36864,     4096;\n  7:   279936,  1306368,   2612736,   2903040,   1935360,    774144,   172032,    16384;\n  8:  1679616,  8957952,  20901888,  27869184,  23224320,  12386304,  4128768,   786432,   65536;\n  9: 10077696, 60466176, 161243136, 250822656, 250822656, 167215104, 74317824, 21233664, 3538944, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(6 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (6 + 4*x)*v[i - 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}},{"oeis_id":"A038236","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      4,       6;\n  2:     16,      48,       36;\n  3:     64,     288,      432,      216;\n  4:    256,    1536,     3456,     3456,      1296;\n  5:   1024,    7680,    23040,    34560,     25920,      7776;\n  6:   4096,   36864,   138240,   276480,    311040,    186624,     46656;\n  7:  16384,  172032,   774144,  1935360,   2903040,   2612736,   1306368,    279936;\n  8:  65536,  786432,  4128768, 12386304,  23224320,  27869184,  20901888,   8957952,  1679616;\n  9: 262144, 3538944, 21233664, 74317824, 167215104, 250822656, 250822656, 161243136, 60466176, 10077696;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (6+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(6 + 4*x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (6 + 4*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A038258","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:        1;\n  1:        6,        4;\n  2:       36,       48,        16;\n  3:      216,      432,       288,        64;\n  4:     1296,     3456,      3456,      1536,       256;\n  5:     7776,    25920,     34560,     23040,      7680,      1024;\n  6:    46656,   186624,    311040,    276480,    138240,     36864,     4096;\n  7:   279936,  1306368,   2612736,   2903040,   1935360,    774144,   172032,    16384;\n  8:  1679616,  8957952,  20901888,  27869184,  23224320,  12386304,  4128768,   786432,   65536;\n  9: 10077696, 60466176, 161243136, 250822656, 250822656, 167215104, 74317824, 21233664, 3538944, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(6 + 4*x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (6 + 4*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-7b91a7a037fb3513","steps":["basis A013611"],"edges":["e-0e13ca82ccc1a33c"],"text":"A038243 \u2192 basis A013611"},"headline":"A038243 \u2192 basis A013611","chain":null}