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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"5e446a6c9c634031fe08f4379109f7051b56a02a1b7ab3596cf89f1e9e90832c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-fb16da45c5e69d3d","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) = 8*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-fb16da45c5e69d3d","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) = 8*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,         42,             4;\n  2:   12,        512,           476,              8;\n  3:   24,       5400,         17464,           4936,              16;\n  4:   48,      54752,        450976,         438912,           49840,              32;\n  5:   96,     549408,       9939712,       23001152,         9555552,          499616,              64;\n  6:  192,    5498624,     201591360,      932508288,       919446848,       192585216,         4999104,            128;\n  7:  384,   54996864,    3899579008,    32636006784,     62785937280,     31918151296,      3712492416,       49997952,           256;\n  8:  768,  549992960,   73342247936,  1041978441728,   3498154623488,   3468004692992,   1013959869440,    69684752384,     499995392,        512;\n  9: 1536, 5499984384, 1356060015616, 31315629918208, 170782370759680, 292083683322880, 168356344508416, 30375012296704, 1286925256192, 4999989760, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(8*n-14)*x)*P(n-2,x) + (8*x-8*x^2)*d/dx P(n-2,x) + (1+(8*n+2)*x)*P(n-1,x) + (8*x-8*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 42*x + 4*x^2.","mathematica":"P[0, x_] := 3 + 2*x;\nP[1, x_] := 6 + 42*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + (8*n - 14)*x)*P[n - 2, x] + (8*x - 8*x^2)*D[P[n - 2, x], x] + (1 + (8*n + 2)*x)*P[n - 1, x] + (8*x - 8*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 + 42*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (8*n - 14)*x)*v[i - 2] + (8*x - 8*x^2)*deriv(v[i - 2], x) + (1 + (8*n + 2)*x)*v[i - 1] + (8*x - 8*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-fb16da45c5e69d3d","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-3a6b6c46f0fb93e5"],"text":"A257618 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257618 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}