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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"3128b8770f9c4d30a73efdd052372a26734cd4b8198cb90df0fc55635417136d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-d55979c30b97975a","role":"sequence","name":"Triangular array A066667 or A008297 unsigned and transposed.","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-d55979c30b97975a","role":"sequence","name":"Triangular array A066667 or A008297 unsigned and transposed.","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,   2;\n  1: 2,   8,    6;\n  2: 2,  18,   42,     24;\n  3: 2,  32,  156,    264,     120;\n  4: 2,  50,  420,   1440,    1920,      720;\n  5: 2,  72,  930,   5400,   14400,    15840,     5040;\n  6: 2,  98, 1806,  15960,   71400,   156240,   146160,     40320;\n  7: 2, 128, 3192,  39984,  270480,   987840,  1834560,   1491840,    362880;\n  8: 2, 162, 5256,  88704,  846720,  4656960, 14394240,  23224320,  16692480,   3628800;\n  9: 2, 200, 8190, 179280, 2298240, 17781120, 82555200, 221356800, 315705600, 203212800, 39916800;\n","recurrence_sentence":"The row polynomials satisfy (2+2*n*x+(n^2+n)*x^2)*P(n,x) = (2+(6*n+2)*x+(5*n^2+5*n)*x^2+(2*n^3+4*n^2+2*n)*x^3)*P(n-1,x) - (2*x^2+(2*n+2)*x^3+(n^2+n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + 2*x.","mathematica":"P[0, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2 + (6*n + 2)*x + (5*n^2 + 5*n)*x^2 + (2*n^3 + 4*n^2 + 2*n)*x^3)*P[n - 1, x] - (2*x^2 + (2*n + 2)*x^3 + (n^2 + n)*x^4)*D[P[n - 1, x], x])/(2 + 2*n*x + (n^2 + n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (6*n + 2)*x + (5*n^2 + 5*n)*x^2 + (2*n^3 + 4*n^2 + 2*n)*x^3)*v[i - 1] - (2*x^2 + (2*n + 2)*x^3 + (n^2 + n)*x^4)*deriv(v[i - 1], x))/(2 + 2*n*x + (n^2 + n)*x^2)); 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-d55979c30b97975a","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-a32213ca02c6584d"],"text":"A089231 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A089231 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}