{"id":"n-e3e0d7324cd0f1b5","revision":1,"first_index":1,"fingerprint":"f6-d146d5cf1a8d0272","name":"obreschkoff_combination of Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-x)^4, 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":[["2","33","24","1"],["2","73","161","51","1"],["2","155","752","664","106","1"],["2","321","2981","5526","2472","217","1"],["2","655","10764","36395","33850","8613","440","1"],["2","1325","36661","207797","346313","184711","28703","887","1"],["2","2667","120144","1079418","2946864","2808348","931368","92766","1782","1"],["2","5353","383549","5250560","22113932","34216442","20385002","4439336","293450","3573","1"],["2","10727","1202396","24362301","151671036","357464142","345361632","136536594","20306514","914299","7156","1"],["2","21477","3721613","109181495","973268433","3338671242","4888196010","3144980646","861094140","90077377","2818041","14323","1"],["2","42979","11415408","476752996","5938142270","28659254211","60600862272","59051337504","26487541998","5186229625","390397232","8618844","28658","1"],["2","85985","34790629","2041185994","34845827656","230495868671","679126532295","952655281932","648734449002","209946468943","30136351027","1662204170","26214764","57329","1"],["2","171999","105547020","8608417631","198335783862","1761182253267","7033197950072","13677342500115","13403813750478","6614959369741","1586081611980","170240618829","6981004090","79418241","114672","1"],["2","344029","319163717","35885142697","1101934748865","12917374161265","68400541030473","179127631515341","242833789519401","172757015714935","63546763913999","11529628333147","940302547667","29009685627","239917475","229359","1"],["2","688091","962880944","148248665630","6005110924100","91660809683654","632255944301192","2179016893494158","3963999695063840","3900293537059780","2074660418809696","581626554548458","81233116099372","5100693519130","119558143720","723307498","458734","1"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["60","288","1680","11520","90720","806400","7983360","87091200","1037836800","13412044800","186810624000","2789705318400","44460928512000","753220435968000","13516122267648000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","33","24","1"],["2","73","161","51","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162804,"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-26f69690cce5954a"]}],"region_exact":{},"incoming_edges":[{"id":"e-26f69690cce5954a","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-e3e0d7324cd0f1b5","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":"79c965e122a6eec10eb5f67358ae4bcbd89acab63affc4c9596f5b7c25486f6a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-6a8ffbe72ba42678","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-x)^4, read by rows.","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-6a8ffbe72ba42678","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-x)^4, read by rows.","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,    33,      24,         1;\n   2: 2,    73,     161,        51,         1;\n   3: 2,   155,     752,       664,       106,          1;\n   4: 2,   321,    2981,      5526,      2472,        217,          1;\n   5: 2,   655,   10764,     36395,     33850,       8613,        440,          1;\n   6: 2,  1325,   36661,    207797,    346313,     184711,      28703,        887,         1;\n   7: 2,  2667,  120144,   1079418,   2946864,    2808348,     931368,      92766,      1782,        1;\n   8: 2,  5353,  383549,   5250560,  22113932,   34216442,   20385002,    4439336,    293450,     3573,       1;\n   9: 2, 10727, 1202396,  24362301, 151671036,  357464142,  345361632,  136536594,  20306514,   914299,    7156,     1;\n  10: 2, 21477, 3721613, 109181495, 973268433, 3338671242, 4888196010, 3144980646, 861094140, 90077377, 2818041, 14323, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+n*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (n+2)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 33*x + 24*x^2 + x^3, P(2,x) = 2 + 73*x + 161*x^2 + 51*x^3 + x^4.","mathematica":"P[1, x_] := 2 + 33*x + 24*x^2 + x^3;\nP[2, x_] := 2 + 73*x + 161*x^2 + 51*x^3 + x^4;\nP[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (n + 2)*x*P[n - 1, x] + (x - 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 + 33*x + 24*x^2 + x^3; v[2] = 2 + 73*x + 161*x^2 + 51*x^3 + x^4; for(n = 3, N, my(i = n); v[i] = (1 + n*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (n + 2)*x*v[i - 1] + (x - 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-6a8ffbe72ba42678","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-26f69690cce5954a"],"text":"A166341 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A166341 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}