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Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"86308adba53afad7026237c8a688daf17c7a158c3bc9190d405004e4f9ebddf2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-61bf3ec06b6c88be","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"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:           1;\n  1:           4,            4;\n  2:          32,           48,            9;\n  3:         384,          704,          216,          16;\n  4:        6144,        12800,         5040,         640,         25;\n  5:      122880,       280576,       129600,       21760,       1500,        36;\n  6:     2949120,      7225344,      3741696,      752640,      70000,      3024,       49;\n  7:    82575360,    214106112,    121024512,    27725824,    3136000,    185472,     5488,     64;\n  8:  2642411520,   7181697024,   4354523136,  1102381056,  143673600,  10450944,   428064,   9216,    81;\n  9: 95126814720, 269110738944, 172921651200, 47427092480, 6894720000, 583123968, 29635200, 890880, 14580, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(16*n-16)*x-9*x^2)*P(n-2,x) + ((16*n-16)*x^2+4*x^3)*d/dx P(n-2,x) + (8*n-4+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (16*n - 16)*x - 9*x^2)*P[n - 2, x] + ((16*n - 16)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (8*n - 4 + 4*x)*P[n - 1, x] - 4*x^2*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] = 1; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 32*n + 16 + (16*n - 16)*x - 9*x^2)*v[i - 2] + ((16*n - 16)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (8*n - 4 + 4*x)*v[i - 1] - 4*x^2*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-9c5c208df8605708","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-2dc9098c1ed2fab8","e-d8ea2708a3a2261a"],"text":"A225471 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A225471 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}