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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":"a01d737e56212bfc6b3b10ae05078189cdda1c9f5214059d31b4cc821dd7291c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-3c0665ea87e1dc8e","role":"sequence","name":"basis_map_A007318 of A Riordan array of coordination sequences.","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,     2;\n  2:    7,    12,     3;\n  3:   17,    46,    27,      4;\n  4:   41,   152,   144,     48,     5;\n  5:   99,   466,   612,    328,    75,     6;\n  6:  239,  1364,  2295,   1712,   625,   108,     7;\n  7:  577,  3870,  7947,   7628,  3875,  1062,   147,    8;\n  8: 1393, 10736, 26040,  30624, 20050,  7632,  1666,  192,   9;\n  9: 3363, 29282, 81936, 114192, 91790, 45036, 13622, 2464, 243, 10;\n","recurrence_sentence":"The row polynomials satisfy (32+(40*n+32)*x+(16*n^2+20*n+10)*x^2+(2*n^3+4*n^2-2*n)*x^3-(n^2+n)*x^4)*P(n,x) = -(64*n-160+(48*n^2-40*n-288)*x+(8*n^3+24*n^2-160*n-154)*x^2+(6*n^3-24*n^2-74*n-12)*x^3-(9*n^2+3*n-10)*x^4+(2*n+2)*x^5)*P(n-1,x) + (64+(64*n+160)*x+(16*n^2+144*n+120)*x^2+(32*n^2+96*n+4)*x^3+(18*n^2+10*n-30)*x^4+(2*n^2-7*n-11)*x^5-(n+1)*x^6)*d/dx 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[(-(64*n - 160 + (48*n^2 - 40*n - 288)*x + (8*n^3 + 24*n^2 - 160*n - 154)*x^2 + (6*n^3 - 24*n^2 - 74*n - 12)*x^3 - (9*n^2 + 3*n - 10)*x^4 + (2*n + 2)*x^5)*P[n - 1, x] + (64 + (64*n + 160)*x + (16*n^2 + 144*n + 120)*x^2 + (32*n^2 + 96*n + 4)*x^3 + (18*n^2 + 10*n - 30)*x^4 + (2*n^2 - 7*n - 11)*x^5 - (n + 1)*x^6)*D[P[n - 1, x], x])/(32 + (40*n + 32)*x + (16*n^2 + 20*n + 10)*x^2 + (2*n^3 + 4*n^2 - 2*n)*x^3 - (n^2 + n)*x^4)]];\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] = (-(64*n - 160 + (48*n^2 - 40*n - 288)*x + (8*n^3 + 24*n^2 - 160*n - 154)*x^2 + (6*n^3 - 24*n^2 - 74*n - 12)*x^3 - (9*n^2 + 3*n - 10)*x^4 + (2*n + 2)*x^5)*v[i - 1] + (64 + (64*n + 160)*x + (16*n^2 + 144*n + 120)*x^2 + (32*n^2 + 96*n + 4)*x^3 + (18*n^2 + 10*n - 30)*x^4 + (2*n^2 - 7*n - 11)*x^5 - (n + 1)*x^6)*deriv(v[i - 1], x))/(32 + (40*n + 32)*x + (16*n^2 + 20*n + 10)*x^2 + (2*n^3 + 4*n^2 - 2*n)*x^3 - (n^2 + n)*x^4)); 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-ec35d1145e9a1211","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-0b1886c9648f8656","e-23216ea9a2efcfa9"],"text":"A113413 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A113413 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}