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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":"26bbde482fe4f5377d4d7d8d4ea7e1f5389f091faaf30699adc01b87f40f0ae8","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-77f76afdd288bbc1","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 5).","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:       25,          30,            3;\n  1:      125,         350,           90,            4;\n  2:      625,        3750,         1875,          200,            5;\n  3:     3125,       38750,        33750,         6500,          375,            6;\n  4:    15625,      393750,       564375,       175000,        17500,          630,           7;\n  5:    78125,     3968750,      9056250,      4252500,       656250,        39900,         980,         8;\n  6:   390625,    39843750,    141796875,     97125000,     21721875,      1984500,       80850,      1440,        9;\n  7:  1953125,   399218750,   2186718750,   2131562500,    664453125,     85601250,     5145000,    150000,     2025,     10;\n  8:  9765625,  3996093750,  33399609375,  45546875000,  19275781250,   3365381250,   279943125,  11880000,   259875,   2750,   11;\n  9: 48828125, 39980468750, 506988281250, 955470312500, 538828125000, 124092375000, 13724287500, 795135000, 25059375, 426250, 3630, 12;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 5*x^2*d/dx P(n-2,x) + (5+2*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 25 + 30*x + 3*x^2, P(1,x) = 125 + 350*x + 90*x^2 + 4*x^3.","mathematica":"P[0, x_] := 25 + 30*x + 3*x^2;\nP[1, x_] := 125 + 350*x + 90*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 5*x^2*D[P[n - 2, x], x] + (5 + 2*x)*P[n - 1, x] + 5*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] = 25 + 30*x + 3*x^2; v[2] = 125 + 350*x + 90*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 5*x^2*deriv(v[i - 2], x) + (5 + 2*x)*v[i - 1] + 5*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-77f76afdd288bbc1","steps":["\u03b8+1"],"edges":["e-2aba5d679ba52759"],"text":"A075500 \u2192 \u03b8+1"},"headline":"A075500 \u2192 \u03b8+1","chain":null}