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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. 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":"769b5e603f4dfa28b38ee679b0193f485b4df1d96c6f60f05e6911abf1f22e76","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-adb5eb667a229c4c","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 6).","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:        36,           72,             9;\n  1:       216,         1008,           324,             16;\n  2:      1296,        12960,          8100,            960,            25;\n  3:      7776,       160704,        174960,          37440,          2250,            36;\n  4:     46656,      1959552,       3510864,        1209600,        126000,          4536,           49;\n  5:    279936,     23701248,      67604544,       35271936,       5670000,        344736,         8232,          64;\n  6:   1679616,    285534720,    1270209600,      966712320,     225212400,      20575296,       814968,       13824,        81;\n  7:  10077696,   3433135104,   23506225920,    25459246080,    8266860000,    1065016512,     62233920,     1728000,     21870,     100;\n  8:  60466176,  41237932032,  430836620544,   652810752000,  287785872000,   50244872832,   4063430448,   164229120,   3367980,   33000,   121;\n  9: 362797056, 495097049088, 7847844516864, 16433389817856, 9653592960000, 2223227077632, 239052933504, 13190335488, 389723400, 6138000, 47916, 144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(24*x-9*x^2)*P(n-2,x) - (72*x-24*x^2-4*x^3)*d/dx P(n-2,x) - (36*x^2-24*x^3)*d^2/dx^2 P(n-2,x) + (6+4*x)*P(n-1,x) + (12*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 36 + 72*x + 9*x^2, P(1,x) = 216 + 1008*x + 324*x^2 + 16*x^3.","mathematica":"P[0, x_] := 36 + 72*x + 9*x^2;\nP[1, x_] := 216 + 1008*x + 324*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[-(24*x - 9*x^2)*P[n - 2, x] - (72*x - 24*x^2 - 4*x^3)*D[P[n - 2, x], x] - (36*x^2 - 24*x^3)*D[P[n - 2, x], {x, 2}] + (6 + 4*x)*P[n - 1, x] + (12*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] = 36 + 72*x + 9*x^2; v[2] = 216 + 1008*x + 324*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(24*x - 9*x^2)*v[i - 2] - (72*x - 24*x^2 - 4*x^3)*deriv(v[i - 2], x) - (36*x^2 - 24*x^3)*deriv(deriv(v[i - 2], x), x) + (6 + 4*x)*v[i - 1] + (12*x - 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-adb5eb667a229c4c","steps":["(\u03b8+1)\u00b2"],"edges":["e-e1cd6a5ad82bff23"],"text":"A075501 \u2192 (\u03b8+1)\u00b2"},"headline":"A075501 \u2192 (\u03b8+1)\u00b2","chain":null}