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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"6fd2876720122a52af80b32d89815b01db66945450b5f77041236e0de59ab07f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-c060a08f0b5770ad","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.","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-c060a08f0b5770ad","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.","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:    3,         2;\n  1:    6,        34,            4;\n  2:   12,       344,          316,             8;\n  3:   24,      2968,         9048,          2664,             16;\n  4:   48,     24320,       184160,        178080,          21680,             32;\n  5:   96,    196000,      3194624,       7192384,        3040864,         174368,             64;\n  6:  192,   1571456,     50866752,     225855360,      221857600,       48021888,        1397184,           128;\n  7:  384,  12579712,    770803840,    6126728064,    11598435200,     5955737216,      724353408,      11182720,          256;\n  8:  768, 100656128,  11336378368,  151538021888,   496301582848,   490829875712,   146353088512,   10624604672,     89473792,       512;\n  9: 1536, 805290496, 163675008000, 3525223348224, 18626838647808, 31519929093120, 18287546793984, 3390206638080, 153151059456, 715817472, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(6*n-10)*x)*P(n-2,x) + (6*x-6*x^2)*d/dx P(n-2,x) + (1+(6*n+2)*x)*P(n-1,x) + (6*x-6*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 34*x + 4*x^2.","mathematica":"P[0, x_] := 3 + 2*x;\nP[1, x_] := 6 + 34*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + (6*n - 10)*x)*P[n - 2, x] + (6*x - 6*x^2)*D[P[n - 2, x], x] + (1 + (6*n + 2)*x)*P[n - 1, x] + (6*x - 6*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3 + 2*x; v[2] = 6 + 34*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (6*n - 10)*x)*v[i - 2] + (6*x - 6*x^2)*deriv(v[i - 2], x) + (1 + (6*n + 2)*x)*v[i - 1] + (6*x - 6*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c060a08f0b5770ad","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-4b3d723d1668996c"],"text":"A257616 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257616 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}