{"id":"n-496cd9fd696bf0bc","revision":1,"first_index":1,"fingerprint":"f6-4ed0ffd3d88a2b71","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+12*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","39","28","1"],["2","85","189","59","1"],["2","179","880","776","122","1"],["2","369","3477","6462","2880","249","1"],["2","751","12524","42515","39530","10013","504","1"],["2","1517","42581","242445","404481","215439","33319","1015","1"],["2","3051","139376","1258082","3440304","3278268","1085224","107574","2038","1"],["2","6121","444573","6114520","25802420","39943722","23782234","5168672","340050","4085","1"],["2","12263","1392892","28352677","176875852","417206678","403065376","159208138","23628578","1058963","8180","1"],["2","24549","4309485","127001615","1134491241","3895517058","5705033250","3669327534","1003635780","104765929","3262785","16371","1"],["2","49123","13214864","554355916","6919181790","33429377491","70719033664","68909517472","30894541198","6042547625","453899472","9976628","32754","1"],["2","98273","40266821","2372747426","40590112976","268787737631","792382451559","1111703711388","756902311578","244812378223","35101921267","1932062818","30339316","65521","1"],["2","196575","122143532","10004498119","230972889126","2053278831947","8204640341240","15959677419171","15640356159342","7716525767293","1849056566700","198243715285","8112702122","91902489","131056","1"],["2","393181","369313701","41697695361","1283000818617","15056696384793","79779675998449","208996834650173","283353992831193","201561118832071","74116429063103","13438612707219","1094761601451","33707136771","277607787","262127","1"],["2","786395","1114100944","172238649990","6990689432580","106822867338054","737323683316360","2542073460958678","4625240913938720","4550883755757700","2420292018125056","678263950638050","94667255449452","5937625526490","138900826920","836886338","524270","1"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["70","336","1960","13440","105840","940800","9313920","101606400","1210809600","15647385600","217945728000","3254656204800","51871083264000","878757175296000","15768809312256000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","39","28","1"],["2","85","189","59","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":162812,"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-e8d26d2660ac53e4"]}],"region_exact":{},"incoming_edges":[{"id":"e-e8d26d2660ac53e4","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-496cd9fd696bf0bc","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":"10eabe4caea51163565b50570c385877894d0f470ec0b74a731ba8ab40a7b10f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-52497a04d5e26922","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+12*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-52497a04d5e26922","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+12*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,    39,      28,         1;\n   2: 2,    85,     189,        59,          1;\n   3: 2,   179,     880,       776,        122,          1;\n   4: 2,   369,    3477,      6462,       2880,        249,          1;\n   5: 2,   751,   12524,     42515,      39530,      10013,        504,          1;\n   6: 2,  1517,   42581,    242445,     404481,     215439,      33319,       1015,          1;\n   7: 2,  3051,  139376,   1258082,    3440304,    3278268,    1085224,     107574,       2038,         1;\n   8: 2,  6121,  444573,   6114520,   25802420,   39943722,   23782234,    5168672,     340050,      4085,       1;\n   9: 2, 12263, 1392892,  28352677,  176875852,  417206678,  403065376,  159208138,   23628578,   1058963,    8180,     1;\n  10: 2, 24549, 4309485, 127001615, 1134491241, 3895517058, 5705033250, 3669327534, 1003635780, 104765929, 3262785, 16371, 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 + 39*x + 28*x^2 + x^3, P(2,x) = 2 + 85*x + 189*x^2 + 59*x^3 + x^4.","mathematica":"P[1, x_] := 2 + 39*x + 28*x^2 + x^3;\nP[2, x_] := 2 + 85*x + 189*x^2 + 59*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 + 39*x + 28*x^2 + x^3; v[2] = 2 + 85*x + 189*x^2 + 59*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-52497a04d5e26922","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-e8d26d2660ac53e4"],"text":"A166343 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A166343 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}