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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"29258e5f6eaad8dbaacca988cf0bb5f4d44654b8b2ab37c4087d6d40be78f14c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-181be2f2eb86cd8e","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 3).","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-181be2f2eb86cd8e","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 3).","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:     4,        1;\n  1:    12,       10,         1;\n  2:    36,       72,        19,         1;\n  3:   108,      468,       243,        31,         1;\n  4:   324,     2916,      2655,       615,        46,        1;\n  5:   972,    17820,     26811,     10035,      1305,       64,       1;\n  6:  2916,   107892,    259119,    147231,     29610,     2457,      85,      1;\n  7:  8748,   650268,   2439963,   2025891,    591381,    73836,    4242,    109,     1;\n  8: 26244,  3910356,  22609935,  26750655,  10896606,  1920429,  162918,   6858,   136,   1;\n  9: 78732, 23488380, 207399771, 343617795, 190199745, 45464328, 5341707, 327510, 10530, 166, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + x.","mathematica":"P[0, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + 3*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] = 4 + x; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-181be2f2eb86cd8e","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-53ebd0bab53d0b02"],"text":"A075498 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A075498 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}