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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"189dfda35dcbfca6ff8db84357bb44a71107986472f77891e17e03a9579e5ecb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-adb5eb667a229c4c","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 6).","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-adb5eb667a229c4c","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 6).","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:        252,           270,             37,              1;\n  1:       1512,          3492,            936,             61,              1;\n  2:       9072,         43416,          20340,           2400,             91,             1;\n  3:      54432,        530064,         409536,          77940,           5130,           127,            1;\n  4:     326592,       6415200,        7901712,        2280096,         231840,          9702,          169,           1;\n  5:    1959552,      77308992,      148646016,       62624016,        9235296,        581112,        16800,         217,         1;\n  6:   11757312,     929667456,     2752937280,     1651622400,      339682896,      30155328,      1286712,       27216,       271,       1;\n  7:   70543872,   11167766784,    50482538496,    42391874880,    11842109280,    1425274704,     84197232,     2593080,     41850,     331,     1;\n  8:  423263232,  134083745280,   919853459712,  1067887535616,   397655153280,   63151998624,   4961558448,   208665072,   4852980,   61710,   397,   1;\n  9: 2539579392, 1609428206592, 16691446020096, 26549154314496, 12997542134016, 2671127103744, 271537453440, 14977481904, 470725992, 8555580, 87912, 469, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 252 + 270*x + 37*x^2 + x^3.","mathematica":"P[0, x_] := 252 + 270*x + 37*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[(6 + x)*P[n - 1, x] + 6*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] = 252 + 270*x + 37*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (6 + x)*v[i - 1] + 6*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-adb5eb667a229c4c","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-288a11b1cf9253fe"],"text":"A075501 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A075501 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}