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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"745ff5da51462f45fe46ad5b7befb1d2b6a458387430a0e453715cb5eecba02e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-c7809537965f400d","role":"sequence","name":"Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","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-c7809537965f400d","role":"sequence","name":"Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","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:         4,          7,           3;\n  2:        18,         51,          48,          15;\n  3:       120,        465,         675,         435,          105;\n  4:      1050,       5145,       10080,        9870,         4830,          945;\n  5:     11340,      67095,      165375,      217350,       160650,        63315,        10395;\n  6:    145530,    1008315,     2993760,     4937625,      4885650,      2900205,       956340,      135135;\n  7:   2162160,   17162145,    59594535,   118243125,    146621475,    116351235,     57702645,    16351335,     2027025;\n  8:  36486450,  326351025,  1297296000,  3008105100,   4483779300,   4455400950,   2951348400,  1256755500,   312161850,   34459425;\n  9: 689188500, 6857425575, 30703347675, 81462080700, 141834993300, 169333614450, 140387697450, 79808028300, 29772943200, 6581750175, 654729075;\n","recurrence_sentence":"The row polynomials satisfy (2*n+(2*n-1)*x)*P(n,x) = (4*n^2+2*n-2+(8*n^2+2*n-3)*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[((4*n^2 + 2*n - 2 + (8*n^2 + 2*n - 3)*x + (4*n^2 - 1)*x^2)*P[n - 1, x])/(2*n + (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] = ((4*n^2 + 2*n - 2 + (8*n^2 + 2*n - 3)*x + (4*n^2 - 1)*x^2)*v[i - 1])/(2*n + (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-c7809537965f400d","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-7d570c2107f82786"],"text":"A085881 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A085881 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}