{"id":"n-f77168e603d1499e","revision":1,"first_index":1,"fingerprint":"f6-314348480e1d44bc","name":"obreschkoff_combination of Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2","1"],["2","11","1"],["2","69","60","1"],["2","367","1121","309","1"],["2","1865","14624","13870","1558","1"],["2","9363","162089","366174","150084","7807","1"],["2","46861","1649196","7461527","7257442","1508475","39056","1"],["2","234359","15983065","131038859","248812025","125430613","14521451","195305","1"],["2","1171857","150487776","2097079116","6953184804","6835411638","1987631568","136200684","976554","1"],["2","5859355","1392280049","31568244428","170135399792","288593712842","165397067270","29731100636","1257251222","4882803","1"],["2","29296853","12743410364","455788397661","3799888288188","10287007037706","10168548724656","3666059465346","427449577074","11492018497","24414052","1"],["2","146484351","115872333041","6391730254385","79520737640985","325570212476886","510261365134458","319201552040418","76310466573300","5977024087675","104409611397","122070301","1"],["2","732421849","1049345136960","87797310418906","1586641786099370","9446612946686751","22158049790964912","21973358638307340","9206932132287318","1516723049780455","81936158664272","945082009914","610351550","1"],["2","3662109347","9479506622041","1188168025819594","30548737943056516","256781080873014881","864427211664104271","1277809819487583708","851900465972483682","249160790526826933","29119812203694967","1107281212292282","8535157033976","3051757799","1"],["2","18310546845","85507209987564","15906900320376275","572468779099974798","6638857923947315757","31075056092468091680","65495951575852820427","65082568968029340990","30479936707368622879","6420245871363545340","544537134922099425","14809098614706034","76975715062935","15258789048","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","14","132","1800","31920","695520","17962560","536215680","18162144000","688085798400","28829030630400","1323453638707200","66060047583129600","3562061389286400000","206340501568481280000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","1"],["2","11","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-11"],["0","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-3"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162834,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-2762452c13f28e99"]}],"region_exact":{},"incoming_edges":[{"id":"e-2762452c13f28e99","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-f77168e603d1499e","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":"25ccc55ce2b12e024642297b16c871ade5cbfe7ee598ed716b4003e2606dade5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-d023ddeb9da11712","role":"sequence","name":"Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-d023ddeb9da11712","role":"sequence","name":"Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"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: 2,       1;\n   2: 2,      11,          1;\n   3: 2,      69,         60,           1;\n   4: 2,     367,       1121,         309,            1;\n   5: 2,    1865,      14624,       13870,         1558,            1;\n   6: 2,    9363,     162089,      366174,       150084,         7807,            1;\n   7: 2,   46861,    1649196,     7461527,      7257442,      1508475,        39056,           1;\n   8: 2,  234359,   15983065,   131038859,    248812025,    125430613,     14521451,      195305,          1;\n   9: 2, 1171857,  150487776,  2097079116,   6953184804,   6835411638,   1987631568,   136200684,     976554,       1;\n  10: 2, 5859355, 1392280049, 31568244428, 170135399792, 288593712842, 165397067270, 29731100636, 1257251222, 4882803, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(4*n-11)*x)*P(n-2,x) + (4*x-4*x^2)*d/dx P(n-2,x) + (4*n-3)*x*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + x, P(2,x) = 2 + 11*x + x^2.","mathematica":"P[1, x_] := 2 + x;\nP[2, x_] := 2 + 11*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (4*n - 11)*x)*P[n - 2, x] + (4*x - 4*x^2)*D[P[n - 2, x], x] + (4*n - 3)*x*P[n - 1, x] + (4*x - 4*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] = 2 + x; v[2] = 2 + 11*x + x^2; for(n = 3, N, my(i = n); v[i] = (1 + (4*n - 11)*x)*v[i - 2] + (4*x - 4*x^2)*deriv(v[i - 2], x) + (4*n - 3)*x*v[i - 1] + (4*x - 4*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}}],"derivation":{"root":"n-d023ddeb9da11712","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-2762452c13f28e99"],"text":"A142459 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A142459 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}