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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"71e62bd1bfd1375424c1adf66a1242e82f5fae2542c5f015fbf880c2515c3290","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-a7e9ebb9860a9dbe","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 2).","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-a7e9ebb9860a9dbe","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 2).","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:    3,      1;\n  1:    6,      7,       1;\n  2:   12,     34,      13,        1;\n  3:   24,    148,     112,       21,        1;\n  4:   48,    616,     820,      280,       31,       1;\n  5:   96,   2512,    5536,     3060,      590,      43,       1;\n  6:  192,  10144,   35728,    30016,     8960,    1106,      57,     1;\n  7:  384,  40768,  224512,   275856,   119616,   22232,    1904,    73,    1;\n  8:  768, 163456, 1387840,  2431360,  1472016,  386400,   48888,  3072,   91,   1;\n  9: 1536, 654592, 8490496, 20838720, 17151520, 6108816, 1070832, 98040, 4710, 111, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3 + x.","mathematica":"P[0, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + 2*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] = 3 + x; for(n = 1, N, my(i = n + 1); v[i] = (2 + x)*v[i - 1] + 2*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-a7e9ebb9860a9dbe","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-b1f98f91a12826e9"],"text":"A075497 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A075497 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}