{"id":"n-91d76a88748fc9ba","revision":0,"first_index":0,"fingerprint":"f6-ca8fcf8c70f1b4a8","name":"Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.","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":false,"rows":[["14","14","2"],["42","72","27","2"],["132","330","220","44","2"],["429","1430","1430","520","65","2"],["1430","6006","8190","4550","1050","90","2"],["4862","24752","43316","33320","11900","1904","119","2"],["16796","100776","217056","217056","108528","27132","3192","152","2"],["58786","406980","1046520","1302336","854658","301644","55860","5040","189","2"],["208012","1634380","4903140","7354710","6056820","2826516","743820","106260","7590","230","2"],["742900","6537520","22472725","39657750","39657750","23377200","8182020","1669800","189750","11000","275","2"],["2674440","26075790","101235420","206220300","244208250","175829940","78146640","21312720","3474900","321750","15444","324","2"],["9694845","103791870","449764770","1041560520","1432145715","1227553470","669574620","232895520","50945895","6792786","522522","21112","377","2"],["35357670","412506150","1975687350","5136787110","8072094030","8072094030","5264409150","2256175350","631729098","113387274","12598586","818090","28210","434","2"],["129644790","1637618400","8597496600","24837212400","44029603800","50538327840","38605667100","19854343080","6872657220","1583822240","237573336","22342320","1241240","36960","495","2"],["477638700","6495886320","37119350400","118107024000","233646504000","303740455200","267291600576","161549868480","67312445200","19232127200","3713790080","472664192","38118080","1832600","47600","560","2"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["30","143","728","3876","21318","120175","690690","4032015","23841480","142498692","859515920","5225264024","31983672534","196947587823","1219199353190"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A286785","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["14","14","2"]],"recurrence":{"denominator":{"coeffs":[["56"],["30"],["4"]]},"terms":[{"coeff":{"coeffs":[["156","84"],["96","66"],["15","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","26","-28"],["1","7","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":170122,"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-5372c1db0d86ac23"]},{"id":170123,"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-5372c1db0d86ac23"]},{"id":170124,"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-5372c1db0d86ac23"]}],"region_exact":{},"incoming_edges":[{"id":"e-5372c1db0d86ac23","operation":{"operation":"theorem_check","theorem":"hypergeometric_row"},"operation_detail":"hypergeometric_row","conclusion_id":"n-91d76a88748fc9ba","premise_count":0,"theorem":"Row-hypergeometric rows: an exact row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, proved for every n by an invariant row-annihilator system of the rule, writes every row as U_0(n) x^v (or its reversal) times the image of (1 + x)^M(n) under a product of nonnegative multiplier sequences: C^k (C > 0); k -> M'!/(M' - k)!, whose exponential generating function is (1 + x)^M' (Polya--Schur, Theorem 7.1 (3) => (1) of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601: a sequence is a multiplier sequence when G(x) = sum_k lambda_k x^k / k! or G(-x) can be written C x^m e^(a x) prod_k (1 + alpha_k x) with a >= 0, alpha_k >= 0); k -> phi(k) for a polynomial phi with only real roots, all <= 0 (Laguerre's theorem, RealRooted isPFMultiplierSequence_eval_of_roots_nonpos); k -> 1/(alpha)_k for alpha > 0 (RealRooted isPFMultiplierSequence_inv_ascPochhammer, via the generalized Laguerre polynomials); products of PF multiplier sequences are PF multiplier sequences (RealRooted IsPFMultiplierSequence.mul) and map real-rooted polynomials with nonnegative coefficients to such polynomials (IsPFMultiplierSequence.diagonalOperator_isPF)","lean_route":null,"lean_declarations":["RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_inv_ascPochhammer","RealRooted.IsPFMultiplierSequence.mul","RealRooted.IsPFMultiplierSequence.diagonalOperator_isPF"],"certificate_hash":"7b5478088726d026d92dde22f218f6adbe692748eba29087432b98b21a247be4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}}],"premises":[]}],"used_by":[{"id":"e-3e298ae40e0eabb3","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-24fe2fab32454333","conclusion_name":"derivative of Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section."},{"id":"e-4e5d17bd5b20b967","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-ed37a889ab35bb0d","conclusion_name":"theta_plus_one of Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section."},{"id":"e-63a5bf569b6939b3","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-664dba754f71e70d","conclusion_name":"shift_x_plus_one of Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section."},{"id":"e-87cea5e82f8c5aeb","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-8f81b8435f04f925","conclusion_name":"mobius_x_over_x_plus_one of Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section."},{"id":"e-f6e13d92215facd5","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-23bacd54faa778ef","conclusion_name":"reversal of Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section."}],"used_by_total":5,"links":[{"id":"l-9b89be716ad9d592","kind":"fingerprint_related","node_a_id":"n-91d76a88748fc9ba","node_b_id":"n-23bacd54faa778ef","fingerprint":"f6-ca8fcf8c70f1b4a8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ca8fcf8c70f1b4a8"},"other_id":"n-23bacd54faa778ef","other":{"id":"n-23bacd54faa778ef","name":"reversal of Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.","first_index":0,"fingerprint":"f6-ca8fcf8c70f1b4a8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2","14","14"],["2","27","72","42"],["2","44","220","330","132"]],"card_rows":[["2","14","14"],["2","27","72","42"],["2","44","220","330","132"],["2","65","520","1430","1430","429"]],"row_sums_preview":["30","143","728","3876","21318","120175","690690","4032015"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-91d76a88748fc9ba","steps":["reverse"],"edges":["e-f6e13d92215facd5"],"text":"A286785 \u2192 reverse"},"headline":"A286785 \u2192 reverse","chain":null}},{"id":"l-9dee7c52596f4222","kind":"oeis_match","node_a_id":"n-91d76a88748fc9ba","node_b_id":null,"fingerprint":null,"oeis_id":"A006013","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A006013","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d8c74e9142452e3d","kind":"oeis_match","node_a_id":"n-91d76a88748fc9ba","node_b_id":null,"fingerprint":null,"oeis_id":"A305574","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A305574","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A286785","relation":"identification"},{"oeis_id":"A006013","relation":"match_row_sums"},{"oeis_id":"A305574","relation":"match_row_sums"}],"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:     14,      14,        2;\n  1:     42,      72,       27,        2;\n  2:    132,     330,      220,       44,        2;\n  3:    429,    1430,     1430,      520,       65,        2;\n  4:   1430,    6006,     8190,     4550,     1050,       90,       2;\n  5:   4862,   24752,    43316,    33320,    11900,     1904,     119,       2;\n  6:  16796,  100776,   217056,   217056,   108528,    27132,    3192,     152,      2;\n  7:  58786,  406980,  1046520,  1302336,   854658,   301644,   55860,    5040,    189,     2;\n  8: 208012, 1634380,  4903140,  7354710,  6056820,  2826516,  743820,  106260,   7590,   230,   2;\n  9: 742900, 6537520, 22472725, 39657750, 39657750, 23377200, 8182020, 1669800, 189750, 11000, 275, 2;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+30*n+56)*P(n,x) = (15*n^2+96*n+156+(12*n^2+66*n+84)*x)*P(n-1,x) + (n+2+(7*n+26)*x-(8*n+28)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 14*x + 2*x^2.","mathematica":"P[0, x_] := 14 + 14*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((15*n^2 + 96*n + 156 + (12*n^2 + 66*n + 84)*x)*P[n - 1, x] + (n + 2 + (7*n + 26)*x - (8*n + 28)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 30*n + 56)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 14 + 14*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15*n^2 + 96*n + 156 + (12*n^2 + 66*n + 84)*x)*v[i - 1] + (n + 2 + (7*n + 26)*x - (8*n + 28)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 30*n + 56)); 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-91d76a88748fc9ba","steps":[],"edges":[],"text":"A286785"},"headline":"A286785: Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.","chain":null}
