{"id":"n-d0e888e78d074358","revision":1,"first_index":1,"fingerprint":"f6-bc5ca063cc7eafc5","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.","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":false,"rows":[["1"],["1","2"],["4","19","4"],["28","222","147","8"],["280","3194","4128","887","16"],["3640","55024","113566","52538","4835","32"],["58240","1107336","3268788","2562676","555684","25167","64"],["1106560","25526192","100544412","117517960","45415640","5301150","128203","128"],["24344320","663605680","3325767376","5352311764","3189383200","695714590","47537320","646519","256"],["608608000","19213911360","118361719296","248493947496","208996478388","72479948400","9696965250","410038434","3245139","512"],["17041024000","613280884160","4523860151424","11916595601160","13329738085440","6669577913796","1460779679100","126758501016","3449054700","16253855","1024"],["528271744000","21396862829440","185167971243328","594284187983424","847260436500600","573395345743416","184961028810432","27042357560232","1584041519676","28535161190","81331739","2048"],["17961239296000","810173435729280","8091418460275200","30924091249343744","54425368966596624","47583361980941184","21085880413010784","4631667155590104","470467413984456","19165101549026","233486520816","406795911","4096"],["664565853952000","33089464522419200","376287064042358400","1681433588198618624","3563641725594203152","3889394416140453792","2254140748277240352","690354303284441712","107436853898083584","7813110294920048","226459713933958","1896367880298","2034278563","8192"],["26582634158080000","1450094174476057600","18566604597800416000","95560293063214856576","239115244553254359872","317262302929956849968","231864418219731739008","93956849710727472528","20683044715166183616","2349904906027064144","125235724104257732","2629491708535292","15325548213172","10172039983","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","27","405","8505","229635","7577955","295540245","13299311025","678264862275","38661097149675","2435649120429525","168059789309637225","12604484198222791875","1020963220056046141875"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A290595","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-5","-4"],["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159888,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0)","region_exact":null,"edges":["e-8066ebfb4deadcc4"]},{"id":159889,"position":1,"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-8066ebfb4deadcc4"]},{"id":159890,"position":2,"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-8066ebfb4deadcc4"]},{"id":159891,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"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 >= 2","region_exact":null,"edges":["e-8066ebfb4deadcc4"]}],"region_exact":{},"incoming_edges":[{"id":"e-8066ebfb4deadcc4","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-d0e888e78d074358","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":"16c03eba3212e335d66e2899dccdf03143bde0ee9c01f6deafc13d7b25f754a3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-35685a58b6434e5d","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-6d0a683f602cfe91","conclusion_name":"obreschkoff_combination of Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0."},{"id":"e-35685a58b6434e5d","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-6d0a683f602cfe91","conclusion_name":"obreschkoff_combination of Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0."}],"used_by_total":2,"links":[{"id":"l-8f71dada815eb22e","kind":"oeis_match","node_a_id":"n-d0e888e78d074358","node_b_id":null,"fingerprint":null,"oeis_id":"A290595","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A290595","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d0ed97050a344146","kind":"oeis_match","node_a_id":"n-d0e888e78d074358","node_b_id":null,"fingerprint":null,"oeis_id":"A011781","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A011781","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A290595","relation":"identification"},{"oeis_id":"A011781","relation":"match_row_sums"},{"oeis_id":"A290595","relation":"match_triangle"}],"bfiles":[{"node_id":"n-d0e888e78d074358","oeis_id":"A011781","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":18,"oeis_terms_available":18,"term_count":200,"raw_size":49303},{"node_id":"n-d0e888e78d074358","oeis_id":"A290595","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":49181}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:         1;\n   2:         1,           2;\n   3:         4,          19,            4;\n   4:        28,         222,          147,            8;\n   5:       280,        3194,         4128,          887,           16;\n   6:      3640,       55024,       113566,        52538,         4835,          32;\n   7:     58240,     1107336,      3268788,      2562676,       555684,       25167,         64;\n   8:   1106560,    25526192,    100544412,    117517960,     45415640,     5301150,     128203,       128;\n   9:  24344320,   663605680,   3325767376,   5352311764,   3189383200,   695714590,   47537320,    646519,     256;\n  10: 608608000, 19213911360, 118361719296, 248493947496, 208996478388, 72479948400, 9696965250, 410038434, 3245139, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-5+(3*n-4)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n - 5 + (3*n - 4)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (3*n - 5 + (3*n - 4)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A290595","target":"triangle","first_row":1,"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:         1,           2;\n  2:         4,          19,            4;\n  3:        28,         222,          147,            8;\n  4:       280,        3194,         4128,          887,           16;\n  5:      3640,       55024,       113566,        52538,         4835,          32;\n  6:     58240,     1107336,      3268788,      2562676,       555684,       25167,         64;\n  7:   1106560,    25526192,    100544412,    117517960,     45415640,     5301150,     128203,       128;\n  8:  24344320,   663605680,   3325767376,   5352311764,   3189383200,   695714590,   47537320,    646519,     256;\n  9: 608608000, 19213911360, 118361719296, 248493947496, 208996478388, 72479948400, 9696965250, 410038434, 3245139, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-2+(3*n-1)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n - 2 + (3*n - 1)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], 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] = (3*n - 2 + (3*n - 1)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x)); 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":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-d0e888e78d074358","steps":[],"edges":[],"text":"A290595"},"headline":"A290595: Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.","chain":null}