{"id":"n-a93042cd70c4761e","revision":1,"first_index":0,"fingerprint":"f6-f9762973ec1089d7","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["144","96","16"],["1728","1728","576","64"],["20736","27648","13824","3072","256"],["248832","414720","276480","92160","15360","1024"],["2985984","5971968","4976640","2211840","552960","73728","4096"],["35831808","83607552","83607552","46448640","15482880","3096576","344064","16384"],["429981696","1146617856","1337720832","891813888","371589120","99090432","16515072","1572864","65536"],["5159780352","15479341056","20639121408","16052649984","8026324992","2675441664","594542592","84934656","7077888","262144"],["61917364224","206391214080","309586821120","275188285440","160526499840","64210599936","17836277760","3397386240","424673280","31457280","1048576"],["743008370688","2724364025856","4540606709760","4540606709760","3027071139840","1412633198592","470877732864","112113745920","18685624320","2076180480","138412032","4194304"],["8916100448256","35664401793024","65384736620544","72649707356160","54487280517120","29059882942464","11301065588736","3228875882496","672682475520","99656663040","9965666304","603979776","16777216"],["106993205379072","463637223309312","927274446618624","1133335434756096","944446195630080","566667717378048","251852318834688","83950772944896","20987693236224","3886609858560","518214647808","47110422528","2617245696","67108864"],["1283918464548864","5991619501228032","12981842252660736","17309123003547648","15866696086585344","10577797391056896","5288898695528448","2014818550677504","587655410614272","130590091247616","21765015207936","2638183661568","219848638464","11274289152","268435456"],["15407021574586368","77035107872931840","179748585036840960","259636845053214720","259636845053214720","190400353039024128","105777973910568960","45333417390243840","15111139130081280","3917702737428480","783540547485696","118718264770560","13190918307840","1014686023680","48318382080","1073741824"],["184884258895036416","986049380773527552","2465123451933818880","3834636480785940480","4154189520851435520","3323351616681148416","2030937099082924032","967112904325201920","362667339121950720","107456989369466880","25073297519542272","4558781367189504","633164078776320","64939905515520","4638564679680","206158430208","4294967296"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["256","4096","65536","1048576","16777216","268435456","4294967296","68719476736","1099511627776","17592186044416","281474976710656","4503599627370496","72057594037927936","1152921504606846976","18446744073709551616"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A038330","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["144","96","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["12","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158635,"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":false,"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-016885d52a801213"]},{"id":158636,"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-016885d52a801213","e-0a35d6a32dc0856a"]},{"id":158637,"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-016885d52a801213","e-0a35d6a32dc0856a"]},{"id":158638,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-016885d52a801213"]},{"id":158639,"position":4,"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-0a35d6a32dc0856a"]},{"id":158640,"position":5,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)","region_exact":null,"edges":["e-19cc8e386ff5c847"]},{"id":158641,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 12]","output_interval":"(-inf, 0]","statement":"an input root r <= 12 gives the root (r - a)/d <= 0; a larger root gives a positive root"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 12] to roots in (-inf, 0]: an input root r <= 12 gives the root (r - a)/d <= 0; a larger root gives a positive root","region_exact":null,"edges":["e-19cc8e386ff5c847"]}],"region_exact":{},"incoming_edges":[{"id":"e-19cc8e386ff5c847","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-a93042cd70c4761e","premise_count":0,"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":"a1012c590260ac6c5a0c1a435485b151e16d5b51664eff0f7ed5855faa683a05","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},{"claim":"basis_map_preserver","input_interval":"(-inf, 12]","output_interval":"(-inf, 0]","statement":"an input root r <= 12 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[]},{"id":"e-0a35d6a32dc0856a","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-a93042cd70c4761e","premise_count":0,"theorem":"polytool parent provers, route periodic_product: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"d9d8bcb8772e4fe2ca60467d6c42465be8b49b7b7e344d80ef0167ed08bc4a05","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},{"id":"e-016885d52a801213","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-a93042cd70c4761e","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"31eb510f38eadf2e7d05b003dc6ae65ad15e790570f917bf9bb34cfd79ec27b2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-1e482bc7f83a739a","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-329498b88a584186","conclusion_name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j."},{"id":"e-6efc8fa5932928ce","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-de403801a492a76b","conclusion_name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j."},{"id":"e-77aaa200c8a95097","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-fcf922cdd7b5dfe1","conclusion_name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j."},{"id":"e-7b68ca495e98f7ea","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-ee879560f84ab2b6","conclusion_name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j."},{"id":"e-dab838a9194ad2c3","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-b05662f2ce3af6a8","conclusion_name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j."},{"id":"e-60ab5e907432f9de","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-0dc20949b68163c6","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j."},{"id":"e-60ab5e907432f9de","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-0dc20949b68163c6","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j."}],"used_by_total":7,"links":[{"id":"l-1fbc2aead3dcea21","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-d4ffddfb9dca0c4b","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-d4ffddfb9dca0c4b","other":{"id":"n-d4ffddfb9dca0c4b","name":"basis_map_A013610 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","6"],["4","24","36"]],"card_rows":[["1"],["2","6"],["4","24","36"],["8","72","216","216"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013610"],"edges":["e-db1885975295b772"],"text":"A130123 \u2192 basis A013610"},"headline":"A130123 \u2192 basis A013610","chain":null}},{"id":"l-252ae92e65e85a26","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-fe3df7f96c918e14","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-fe3df7f96c918e14","other":{"id":"n-fe3df7f96c918e14","name":"derivative of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["6","18"]],"card_rows":[["3"],["6","18"],["9","54","81"],["12","108","324","324"]],"row_sums_preview":["0","3","24","144","768","3840","18432","86016"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["D"],"edges":["e-5fa4095a1ef98485"],"text":"A013610 \u2192 D"},"headline":"A013610 \u2192 D","chain":null}},{"id":"l-377c14115b7e9c30","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-3e79e2d28a06cec2","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-3e79e2d28a06cec2","other":{"id":"n-3e79e2d28a06cec2","name":"basis_map_A013610 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","3"],["1","6","9"]],"card_rows":[["1"],["1","3"],["1","6","9"],["1","9","27","27"]],"row_sums_preview":["1","4","16","64","256","1024","4096","16384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013610"],"edges":["e-41e3c7da1a12c920"],"text":"basis-map coefficients of A026729 \u2192 basis A013610"},"headline":"basis-map coefficients of A026729 \u2192 basis A013610","chain":null}},{"id":"l-4056661e76e01fb6","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-af427f3533ba7386","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-af427f3533ba7386","other":{"id":"n-af427f3533ba7386","name":"basis_map_A013610 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-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-3"],["2","12","18"]],"card_rows":[["1"],["-1","-3"],["2","12","18"],["-6","-54","-162","-162"]],"row_sums_preview":["1","-4","32","-384","6144","-122880","2949120","-82575360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["basis A013610"],"edges":["e-46ba877ad14f98f3"],"text":"basis-map coefficients of A021012 \u2192 basis A013610"},"headline":"basis-map coefficients of A021012 \u2192 basis A013610","chain":null}},{"id":"l-42bcd38dc47484b9","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-7a4a43fcd9fa4a51","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-7a4a43fcd9fa4a51","other":{"id":"n-7a4a43fcd9fa4a51","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","3"],["81","54","9"]],"card_rows":[["1"],["9","3"],["81","54","9"],["729","729","243","27"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-366b2525683554ec","steps":["basis A013610"],"edges":["e-b4f43a8046f9cfa3"],"text":"A038279 \u2192 basis A013610"},"headline":"A038279 \u2192 basis A013610","chain":null}},{"id":"l-478de156ec4a640c","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-ff3307aeb79ab12b","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-ff3307aeb79ab12b","other":{"id":"n-ff3307aeb79ab12b","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","2"],["36","24","4"]],"card_rows":[["1"],["6","2"],["36","24","4"],["216","216","72","8"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["basis A013609"],"edges":["e-037f2552a98ec0cc"],"text":"A038243 \u2192 basis A013609"},"headline":"A038243 \u2192 basis A013609","chain":null}},{"id":"l-48e8b17924277a7f","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-a93042cd70c4761e","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-8d887bfc40d036b9","other":{"id":"n-8d887bfc40d036b9","name":"Cube of lower triangular normalized binomial matrix.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","1"],["9","6","1"]],"card_rows":[["1"],["3","1"],["9","6","1"],["27","27","9","1"]],"row_sums_preview":["1","4","16","64","256","1024","4096","16384"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A027465"],"derivation":{"root":"n-8d887bfc40d036b9","steps":[],"edges":[],"text":"A027465"},"headline":"A027465: Cube of lower triangular normalized binomial matrix.","chain":null}},{"id":"l-5c1a3750067f9c02","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-4575a0f9ca60b9ab","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-4575a0f9ca60b9ab","other":{"id":"n-4575a0f9ca60b9ab","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 runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["3","9"],["3","18","27"]],"card_rows":[["3"],["3","9"],["3","18","27"],["3","27","81","81"]],"row_sums_preview":["3","12","48","192","768","3072","12288","49152"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["x \u21a6 x/(x+1)"],"edges":["e-e61ce0e3d93770a8"],"text":"A120909 \u2192 x \u21a6 x/(x+1)"},"headline":"A120909 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7061503658212849","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-b8d8f9de0e27e805","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-b8d8f9de0e27e805","other":{"id":"n-b8d8f9de0e27e805","name":"basis_map_A013610 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","6","9"],["1","9","27","27"],["1","12","54","108","81"]],"card_rows":[["1","6","9"],["1","9","27","27"],["1","12","54","108","81"],["1","15","90","270","405","243"]],"row_sums_preview":["16","64","256","1024","4096","16384","65536","262144"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013610"],"edges":["e-33def10999a9e794"],"text":"basis-map coefficients of A112227 \u2192 basis A013610"},"headline":"basis-map coefficients of A112227 \u2192 basis A013610","chain":null}},{"id":"l-840fd4c34071362b","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-63e0d27d213f2270","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-63e0d27d213f2270","other":{"id":"n-63e0d27d213f2270","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","12"],["16","96","144"]],"card_rows":[["1"],["4","12"],["16","96","144"],["64","576","1728","1728"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["basis A013611"],"edges":["e-82b80af703a32f5e"],"text":"A013610 \u2192 basis A013611"},"headline":"A013610 \u2192 basis A013611","chain":null}},{"id":"l-a0fbe08d2676eea2","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-a22bdff1819212bb","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-a22bdff1819212bb","other":{"id":"n-a22bdff1819212bb","name":"basis_map_A007318 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-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","3"],["27","18","3"]],"card_rows":[["3"],["9","3"],["27","18","3"],["81","81","27","3"]],"row_sums_preview":["3","12","48","192","768","3072","12288","49152"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A007318"],"edges":["e-18f137fab6db112d"],"text":"A120910 \u2192 basis A007318"},"headline":"A120910 \u2192 basis A007318","chain":null}},{"id":"l-c4dfca2c743f7c48","kind":"fingerprint_related","node_a_id":"n-6a5af5e1168bb679","node_b_id":"n-a93042cd70c4761e","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-6a5af5e1168bb679","other":{"id":"n-6a5af5e1168bb679","name":"Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","3"],["1","6","9"]],"card_rows":[["1"],["1","3"],["1","6","9"],["1","9","27","27"]],"row_sums_preview":["1","4","16","64","256","1024","4096","16384"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A013610"],"derivation":{"root":"n-6a5af5e1168bb679","steps":[],"edges":[],"text":"A013610"},"headline":"A013610: Triangle of coefficients in expansion of (1+3*x)^n.","chain":null}},{"id":"l-da1f2fcb1b45f137","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-96b6c636cb527859","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-96b6c636cb527859","other":{"id":"n-96b6c636cb527859","name":"basis_map_A013610 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-3"],["1","6","9"]],"card_rows":[["1"],["-1","-3"],["1","6","9"],["-1","-9","-27","-27"]],"row_sums_preview":["1","-4","16","-64","256","-1024","4096","-16384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013610"],"edges":["e-226103a223be7e17"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013610"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013610","chain":null}},{"id":"l-e15acd3f400f0683","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-1cfea80692e421ba","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-1cfea80692e421ba","other":{"id":"n-1cfea80692e421ba","name":"basis_map_A013610 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-3"],["1","6","9"]],"card_rows":[["1"],["-1","-3"],["1","6","9"],["-1","-9","-27","-27"]],"row_sums_preview":["1","-4","16","-64","256","-1024","4096","-16384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013610"],"edges":["e-fe4cc1f9dd1d3096"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013610"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013610","chain":null}},{"id":"l-ea5970505bb39274","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-412b54b3d1f71393","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-412b54b3d1f71393","other":{"id":"n-412b54b3d1f71393","name":"basis_map_A013610 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","3"],["2","12","18"]],"card_rows":[["1"],["1","3"],["2","12","18"],["6","54","162","162"]],"row_sums_preview":["1","4","32","384","6144","122880","2949120","82575360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013610"],"edges":["e-77bf2cb365051314"],"text":"A136572 \u2192 basis A013610"},"headline":"A136572 \u2192 basis A013610","chain":null}},{"id":"l-fcf71fa884d71349","kind":"fingerprint_related","node_a_id":"n-a93042cd70c4761e","node_b_id":"n-f5aba8978740eb12","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-f5aba8978740eb12","other":{"id":"n-f5aba8978740eb12","name":"derivative of Cube of lower triangular normalized binomial matrix.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["6","2"]],"card_rows":[["1"],["6","2"],["27","18","3"],["108","108","36","4"]],"row_sums_preview":["0","1","8","48","256","1280","6144","28672"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["D"],"edges":["e-6602f7f73d905d9e"],"text":"A027465 \u2192 D"},"headline":"A027465 \u2192 D","chain":null}},{"id":"l-958dddbc4b887508","kind":"oeis_match","node_a_id":"n-a93042cd70c4761e","node_b_id":null,"fingerprint":null,"oeis_id":"A001025","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001025","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001025","relation":"match_row_sums"},{"oeis_id":"A038330","relation":"tail"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:          144,            96,            16;\n  1:         1728,          1728,           576,            64;\n  2:        20736,         27648,         13824,          3072,           256;\n  3:       248832,        414720,        276480,         92160,         15360,          1024;\n  4:      2985984,       5971968,       4976640,       2211840,        552960,         73728,         4096;\n  5:     35831808,      83607552,      83607552,      46448640,      15482880,       3096576,       344064,        16384;\n  6:    429981696,    1146617856,    1337720832,     891813888,     371589120,      99090432,     16515072,      1572864,       65536;\n  7:   5159780352,   15479341056,   20639121408,   16052649984,    8026324992,    2675441664,    594542592,     84934656,     7077888,     262144;\n  8:  61917364224,  206391214080,  309586821120,  275188285440,  160526499840,   64210599936,  17836277760,   3397386240,   424673280,   31457280,   1048576;\n  9: 743008370688, 2724364025856, 4540606709760, 4540606709760, 3027071139840, 1412633198592, 470877732864, 112113745920, 18685624320, 2076180480, 138412032, 4194304;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 96*x + 16*x^2.","mathematica":"P[0, x_] := 144 + 96*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(12 + 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] = 144 + 96*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 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}}],"derivation":{"root":"n-a93042cd70c4761e","steps":[],"edges":[],"text":"A038330"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","chain":null}
