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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"7334aadff4ce6661c0d07c445ffc23b75c6b9a68d7279b76da1225a316fa7cf2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-58f68e861f34b78a","role":"sequence","name":"Triangle read by rows, S_3(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-58f68e861f34b78a","role":"sequence","name":"Triangle read by rows, S_3(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:    3,       1;\n   2:    6,       8,        1;\n   3:   12,      46,       16,         1;\n   4:   24,     242,      174,        27,         1;\n   5:   48,    1234,     1634,       471,        41,        1;\n   6:   96,    6218,    14306,      6815,      1045,       58,       1;\n   7:  192,   31186,   120666,     89271,     21445,     2031,      78,      1;\n   8:  384,  156122,   996514,   1102647,    389501,    55972,    3591,    101,    1;\n   9:  768,  780994,  8128234,  13125631,   6555661,  1341025,  127792,   5914,  127,   1;\n  10: 1536, 3905738, 65806866, 152510175, 104904885, 29353086, 3896865, 263814, 9216, 156, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 3 + x.","mathematica":"P[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + 3*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] = 3 + x; for(n = 2, N, my(i = n); v[i] = (2 + x)*v[i - 1] + 3*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-58f68e861f34b78a","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-dfae76bc29223b34"],"text":"A225468 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A225468 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}