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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":"0abaf3328d67f3b3d787cbaf20a11b3151d1499df2f1f51977ee7315ced83325","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-a5324901f71bbafe","role":"sequence","name":"basis_map_A008277 of Triangle read by rows, s_3(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:          3,           4;\n  2:         18,          40,           9;\n  3:        162,         472,         189,          16;\n  4:       1944,        6784,        3708,         576,         25;\n  5:      29160,      116320,       78300,       16960,       1375,        36;\n  6:     524880,     2326720,     1827720,      495360,      56750,      2808,       49;\n  7:   11022480,    53287360,    47272680,    15137920,    2192750,    154728,     5145,     64;\n  8:  264539520,  1376273920,  1348986240,   492764160,   85050000,   7644672,   364952,   8704,    81;\n  9: 7142567040, 39593827840, 42217217280, 17181122560, 3406494000, 367101504, 22448664, 772608, 13851, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-21*n+12+(12*n-12)*x-9*x^2)*P(n-2,x) - ((6*n-6)*x-(12*n-12)*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (6*n-4+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 21*n + 12 + (12*n - 12)*x - 9*x^2)*P[n - 2, x] - ((6*n - 6)*x - (12*n - 12)*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (6*n - 4 + 4*x)*P[n - 1, x] + (2*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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 21*n + 12 + (12*n - 12)*x - 9*x^2)*v[i - 2] - ((6*n - 6)*x - (12*n - 12)*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (6*n - 4 + 4*x)*v[i - 1] + (2*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-8229dbdd6a4b5cd9","steps":["basis A008277","(\u03b8+1)\u00b2"],"edges":["e-30984bb1804f505f","e-6596990b8f3f2407"],"text":"A225470 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2"},"headline":"A225470 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2","chain":null}