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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"db095e70c5adca13d87ba448931b7f5337f4fbcdd46d702192cbbc65d4d67734","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-f425461cbf5769b3","role":"sequence","name":"Triangle read by rows, 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-f425461cbf5769b3","role":"sequence","name":"Triangle read by rows, 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,          32;\n   3:    36,       356,         704,          384;\n   4:   108,      2636,       10592,        14208,          6144;\n   5:   324,     18884,      137600,       340224,        344064,         122880;\n   6:   972,    133484,     1664672,      6754560,      11980800,        9707520,        2949120;\n   7:  2916,    938276,    19379264,    121294464,     335708160,      462888960,      312606720,       82575360;\n   8:  8748,   6579596,   220678112,   2051968128,    8319166464,    17360609280,    19549716480,    11312824320,     2642411520;\n   9: 26244,  46092164,  2480096000,  33427659264,  190895652864,   565677342720,   944496967680,   898089615360,   454494781440,    95126814720;\n  10: 78732, 322750124, 27649793312, 531176040960, 4161859952640, 16828491939840, 39077674352640, 54286693171200, 44646185041920, 20071757905920, 3805072588800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+4*x)*P(n-1,x) + (4*x+4*x^2)*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 + 4*x^2)*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 + 4*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f425461cbf5769b3","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-d4e2c570702f66e5"],"text":"A225473 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A225473 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}