{"id":"n-6ec7c64ebc7ea311","revision":1,"first_index":0,"fingerprint":"f6-61bffa5b2969c8c2","name":"obreschkoff_combination of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 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":true,"rows":[["4","1"],["24","11","1"],["252","141","22","1"],["3696","2327","467","37","1"],["69300","47385","11116","1166","56","1"],["1580040","1150371","301269","37790","2450","79","1"],["42397740","32459301","9241954","1317079","103720","4579","106","1"],["1308273120","1044215343","317797479","49924763","4522819","245357","7861","137","1"],["45626025060","37730393649","12131321688","2060042508","207818128","13092310","520072","12652","172","1"],["1774345419000","1513059932475","509616554985","92278402124","10147263524","717454794","33344094","1012956","19356","211","1"],["76119418475100","66689052385725","23379281894574","4469542253661","527797147120","40946413850","2149201572","76851138","1844580","28425","254","1"],["3570692721195600","3204599511003975","1163524125113019","233086699800801","29236232700741","2449460184710","141827560622","5757203298","163469658","3179715","40359","301","1"],["181807771054209300","166744416942455625","62452803940507332","13033428713955066","1722063154417672","153997734878911","9674200297256","435141526028","14086854276","325522263","5237012","55706","352","1"],["9985442194823495400","9340193155685195475","3597086063121167421","778376553509831562","107632028689244982","10182301699260977","685519906464679","33584258818396","1209271934156","31977972741","613397499","8299642","75062","407","1"],["588427843623527407500","560402418259684279125","221323022234621235450","49470083835043613715","7121892867075296856","707781474879623781","50589838224251326","2665273336056331","104875602093548","3094682842823","68147637558","1102854753","12726896","99071","466","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["5","36","416","6528","129024","3072000","85524480","2724986880","97769226240","3900199403520","171228266496000","8203736501452800","425924605299916800","23819632643565158400","1427506405445940019200"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4","1"]],"recurrence":{"denominator":{"coeffs":[["0","1"],["4","0"]]},"terms":[{"coeff":{"coeffs":[["-4","3","1"],["12","8","0"],["16","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162851,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-6a5312030eb510ce"]}],"region_exact":{},"incoming_edges":[{"id":"e-6a5312030eb510ce","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-6ec7c64ebc7ea311","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"cbb692bced09b680b362c3bde9aaa7c96ddd90998b2816cf65cd679ad0d3086a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-9c5c208df8605708","role":"sequence","name":"Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-9c5c208df8605708","role":"sequence","name":"Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"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:             4,             1;\n  1:            24,            11,            1;\n  2:           252,           141,           22,           1;\n  3:          3696,          2327,          467,          37,           1;\n  4:         69300,         47385,        11116,        1166,          56,         1;\n  5:       1580040,       1150371,       301269,       37790,        2450,        79,        1;\n  6:      42397740,      32459301,      9241954,     1317079,      103720,      4579,      106,       1;\n  7:    1308273120,    1044215343,    317797479,    49924763,     4522819,    245357,     7861,     137,     1;\n  8:   45626025060,   37730393649,  12131321688,  2060042508,   207818128,  13092310,   520072,   12652,   172,   1;\n  9: 1774345419000, 1513059932475, 509616554985, 92278402124, 10147263524, 717454794, 33344094, 1012956, 19356, 211, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n+x)*P(n,x) = (16*n^2+12*n-4+(8*n+3)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 4 + x.","mathematica":"P[0, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^2 + 12*n - 4 + (8*n + 3)*x + x^2)*P[n - 1, x])/(4*n + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4 + x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^2 + 12*n - 4 + (8*n + 3)*x + x^2)*v[i - 1])/(4*n + 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-9c5c208df8605708","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-6a5312030eb510ce"],"text":"A225471 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A225471 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}