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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"4fc3f3810a2eebd722eddc81f23e694422af434fd5e4b48f79c22e1ea811a90f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-d55b2cfe05a04774","role":"sequence","name":"Triangle of B-analogs of Stirling numbers of the second kind.","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-d55b2cfe05a04774","role":"sequence","name":"Triangle of B-analogs of Stirling numbers of the second kind.","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: 2,     1;\n   2: 2,     5,       1;\n   3: 2,    17,      10,       1;\n   4: 2,    53,      67,      17,       1;\n   5: 2,   161,     388,     186,      26,       1;\n   6: 2,   485,    2101,    1690,     420,      37,      1;\n   7: 2,  1457,   10990,   13931,    5470,     827,     50,     1;\n   8: 2,  4373,   56407,  108507,   63161,   14567,   1477,    65,    1;\n   9: 2, 13121,  286408,  815956,  676956,  223398,  33768,  2452,   82,   1;\n  10: 2, 39365, 1445161, 5998100, 6908560, 3134334, 662382, 70548, 3846, 101, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2 + x.","mathematica":"P[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 2*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] = 2 + x; for(n = 2, N, my(i = n); v[i] = (1 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-d55b2cfe05a04774","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-a691d30fc1b4b246"],"text":"A039755 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A039755 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}