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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":"1381a217dd6b4dcb1a0c7ff7a2d81a346c2c4ec7744959a941c0652ca2276ec2","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-d7b6e6dc2dbf6491","role":"sequence","name":"basis_map_A008277 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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:         2,         2;\n  2:         8,        14,          3;\n  3:        48,       114,         45,         4;\n  4:       384,      1122,        621,       104,         5;\n  5:      3840,     13110,       9135,      2180,       200,        6;\n  6:     46080,    178110,     147555,     44880,      5925,      342,       7;\n  7:    645120,   2763810,    2628045,    959700,    162750,    13608,     539,     8;\n  8:  10321920,  48275010,   51450525,  21738360,   4454625,   481068,   27734,   800,    9;\n  9: 185794560, 937869030, 1101423015, 525106260, 125174700, 16410114, 1226862, 51696, 1134, 10;\n","recurrence_sentence":"The row polynomials satisfy (2-2*x+2*n*x^2)*P(n,x) = (4*n-(4*n-4)*x+(4*n^2-4)*x^2+(2*n+2)*x^3)*P(n-1,x) + (2*x-3*x^2+(2*n+3)*x^3)*d/dx P(n-1,x) + x^3*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - (4*n - 4)*x + (4*n^2 - 4)*x^2 + (2*n + 2)*x^3)*P[n - 1, x] + (2*x - 3*x^2 + (2*n + 3)*x^3)*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}])/(2 - 2*x + 2*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n - (4*n - 4)*x + (4*n^2 - 4)*x^2 + (2*n + 2)*x^3)*v[i - 1] + (2*x - 3*x^2 + (2*n + 3)*x^3)*deriv(v[i - 1], x) + x^3*deriv(deriv(v[i - 1], x), x))/(2 - 2*x + 2*n*x^2)); 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-4f6e747b794bcc2e","steps":["basis A008277","\u03b8+1"],"edges":["e-760cba6118ea84eb","e-adda163aee77ed9d"],"text":"A028338 \u2192 basis A008277 \u2192 \u03b8+1"},"headline":"A028338 \u2192 basis A008277 \u2192 \u03b8+1","chain":null}