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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"273822f9576f7e0c49356a969085ff6bc75ae09674b5541d16ebf366304aacd0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-4c2a0f8cb422ddaa","role":"sequence","name":"Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).","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-4c2a0f8cb422ddaa","role":"sequence","name":"Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).","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,        4;\n  1: 2,       28,         16;\n  2: 2,      148,        256,          64;\n  3: 2,      748,       2896,        1856,         256;\n  4: 2,     3748,      29056,       35712,       11776,        1024;\n  5: 2,    18748,     276496,      580480,      343040,       68608,        4096;\n  6: 2,    93748,    2563456,     8652224,     8153600,     2812928,      376832,      16384;\n  7: 2,   468748,   23446096,   122732736,   173220096,    91685888,    20672512,    1982464,     65536;\n  8: 2,  2343748,  212889856,  1689309952,  3435672576,  2618284032,   883556352,  140181504,  10092544,   262144;\n  9: 2, 11718748, 1925383696, 22812588800, 65163673600, 68726654976, 32562044928, 7599489024, 893779968, 50069504, 1048576;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + 4*x.","mathematica":"P[0, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 4*x)*P[n - 1, x] + 4*x*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] = 2 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = (1 + 4*x)*v[i - 1] + 4*x*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-4c2a0f8cb422ddaa","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-f7eb819d34bc450b"],"text":"A285061 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A285061 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}