{"id":"n-02156bdfcfec6826","revision":1,"first_index":0,"fingerprint":"f6-c64977f436170997","name":"Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.","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":[["1","12","9"],["1","33","99","27"],["1","78","594","702","81"],["1","171","2718","8154","4617","243"],["1","360","10719","65232","96471","29160","729"],["1","741","38637","421713","1265139","1043199","180063","2187"],["1","1506","131472","2382318","12651390","21440862","10649232","1097874","6561"],["1","3039","430560","12290184","106138674","318416022","331834968","104626080","6646293","19683"],["1","6108","1373733","59494176","788787234","3820992264","7099085106","4819028256","1001451357","40074588","59049"],["1","12249","4304439","275067495","5371796394","39490485498","118471456494","145038502638","66841401285","9413808093","241097067","177147"],["1","24534","13317534","1229403150","34285804335","365866026444","1658600390916","3292794237996","2777150151135","896234896350","87376340574","1448708166","531441"],["1","49107","40835826","5357091222","208311116175","3120912875709","20390987371068","61172962113204","84264647644143","50619601230525","11715958502514","803771563158","8699157729","1594323"],["1","98256","124422651","22898454624","1218339591201","24974810739504","227001559241619","978767393811264","2043014033174571","2022959669899824","888169561985529","150236760788064","7347033118899","52217266896","4782969"],["1","196557","377394705","96446301885","6916042322469","190054070946657","2338255236876453","13959181250013825","41877543750041475","63132891395664231","46183139240037651","15125384559239703","1898352560002455","66854339806635","313375345911","14348907"],["1","393162","1141029180","401635785150","38341617385860","1389301949288826","22639570999374852","181821107106404190","753795787500746550","1636389963957637710","1833805250949363012","1012801121031554154","251559351668627460","23716191477322350","606389688448380","1880481657978","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["22","160","1456","15904","202672","2951680","48361216","880405504","17630351872","385148108800","9114999832576","232311251144704","6343764407375872","184778982658539520","5718564661248065536"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A156366","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","12","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","6"],["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159450,"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-e495670ea063ae0d"]},{"id":159451,"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-e495670ea063ae0d"]},{"id":159452,"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-e495670ea063ae0d"]},{"id":159453,"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-e495670ea063ae0d"]}],"region_exact":{},"incoming_edges":[{"id":"e-e495670ea063ae0d","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-02156bdfcfec6826","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":"e49d80a6782ef8ad469747347dcd81a59549d27aa64a4fe6c503d1656439e80f","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-4796775b805410d0","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-9aad4442f558f61d","conclusion_name":"obreschkoff_combination of Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows."},{"id":"e-4796775b805410d0","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-9aad4442f558f61d","conclusion_name":"obreschkoff_combination of Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows."}],"used_by_total":2,"links":[{"id":"l-52757525223b4085","kind":"oeis_match","node_a_id":"n-02156bdfcfec6826","node_b_id":null,"fingerprint":null,"oeis_id":"A122704","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":"A122704","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A156366","relation":"identification"},{"oeis_id":"A122704","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: 1,    12,       9;\n  1: 1,    33,      99,        27;\n  2: 1,    78,     594,       702,         81;\n  3: 1,   171,    2718,      8154,       4617,         243;\n  4: 1,   360,   10719,     65232,      96471,       29160,          729;\n  5: 1,   741,   38637,    421713,    1265139,     1043199,       180063,         2187;\n  6: 1,  1506,  131472,   2382318,   12651390,    21440862,     10649232,      1097874,        6561;\n  7: 1,  3039,  430560,  12290184,  106138674,   318416022,    331834968,    104626080,     6646293,      19683;\n  8: 1,  6108, 1373733,  59494176,  788787234,  3820992264,   7099085106,   4819028256,  1001451357,   40074588,     59049;\n  9: 1, 12249, 4304439, 275067495, 5371796394, 39490485498, 118471456494, 145038502638, 66841401285, 9413808093, 241097067, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(3*n+6)*x)*P(n-1,x) + (x-3*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 12*x + 9*x^2.","mathematica":"P[0, x_] := 1 + 12*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (3*n + 6)*x)*P[n - 1, x] + (x - 3*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 12*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (3*n + 6)*x)*v[i - 1] + (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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-02156bdfcfec6826","steps":[],"edges":[],"text":"A156366"},"headline":"A156366: Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.","chain":null}