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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"1c8f145c763ff763353f9ae30d8e7c15a0cae9f7df24218e52947797db16d5ae","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-de5746895014fee3","role":"sequence","name":"Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-de5746895014fee3","role":"sequence","name":"Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     4,         4;\n   2:    12,        44,          16;\n   3:    36,       356,         352,          64;\n   4:   108,      2636,        5296,        2368,          256;\n   5:   324,     18884,       68800,       56704,        14336,         1024;\n   6:   972,    133484,      832336,     1125760,       499200,        80896,        4096;\n   7:  2916,    938276,     9689632,    20215744,     13987840,      3857408,      434176,       16384;\n   8:  8748,   6579596,   110339056,   341994688,    346631936,    144671744,    27152384,     2244608,      65536;\n   9: 26244,  46092164,  1240048000,  5571276544,   7953985536,   4713977856,  1311801344,   178192384,   11272192,   262144;\n  10: 78732, 322750124, 13824896656, 88529340160, 173410831360, 140237432832, 54274547712, 10771169280, 1107296256, 55312384, 1048576;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + 4*x.","mathematica":"P[1, x_] := 4 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 4 + 4*x; for(n = 2, N, my(i = n); v[i] = (3 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-de5746895014fee3","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-9a50147d814dfb7b"],"text":"A225467 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A225467 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}