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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"e76426ec15730a25edd794617aa55c2abf51465a6257dadd45b9811ee0231879","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-2b477f2aa77e3efc","role":"sequence","name":"Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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-2b477f2aa77e3efc","role":"sequence","name":"Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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: 2,   1;\n  1: 2,   5,    3;\n  2: 2,  13,   26,     15;\n  3: 2,  25,  109,    191,     105;\n  4: 2,  41,  316,   1126,    1794,      945;\n  5: 2,  61,  735,   4430,   13828,    20469,     10395;\n  6: 2,  85, 1478,  13525,   69518,   196855,    274602,    135135;\n  7: 2, 113, 2681,  34685,  265433,  1215347,   3188379,   4233375,    2027025;\n  8: 2, 145, 4504,  78316,  834988,  5612950,  23494616,  57907404,   73724130,   34459425;\n  9: 2, 181, 7131, 160476, 2273712, 21061614, 128064634, 498456764, 1165768326, 1431163845, 654729075;\n","recurrence_sentence":"The row polynomials satisfy (2+(2*n-1)*x)*P(n,x) = (2+(6*n-1)*x+(4*n^2-1)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2 + x.","mathematica":"P[0, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((2 + (6*n - 1)*x + (4*n^2 - 1)*x^2)*P[n - 1, x])/(2 + (2*n - 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (6*n - 1)*x + (4*n^2 - 1)*x^2)*v[i - 1])/(2 + (2*n - 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-2b477f2aa77e3efc","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-fbfe6dbd1940a725"],"text":"A109692 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A109692 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}