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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":"f540186bf8585aabd89f6dfb62a808184a3595e2afbc7d070fe7c9cd5dfd3fa6","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-9fa98aec39d32ffa","role":"sequence","name":"Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).","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,   4;\n  2:  3,  20,     9;\n  3:  4,  52,    81,    16;\n  4:  5, 104,   315,   224,     25;\n  5:  6, 180,   864,  1200,    500,     36;\n  6:  7, 284,  1944,  4272,   3500,    972,     49;\n  7:  8, 420,  3843, 12000,  15575,   8568,   1715,    64;\n  8:  9, 592,  6930, 28800,  53075,  46368,  18522,  2816,   81;\n  9: 10, 804, 11664, 61728, 151150, 187596, 119364, 36480, 4374, 100;\n","recurrence_sentence":"The row polynomials satisfy 24*P(n,x) = -(152+876*x-540*x^2-368*x^3+24*x^4)*P(n-4,x) - (27+119*x-45*x^2-432*x^3-128*x^4)*d/dx P(n-4,x) - (27*x-9*x^2-9*x^3-36*x^4)*d^2/dx^2 P(n-4,x) + (568+1400*x-1344*x^2+96*x^3)*P(n-3,x) + (92+396*x-650*x^2-1056*x^3)*d/dx P(n-3,x) + (92*x-76*x^2-130*x^3-96*x^4)*d^2/dx^2 P(n-3,x) - (596+336*x+144*x^2)*P(n-2,x) - (103+300*x-960*x^2)*d/dx P(n-2,x) - (103*x-152*x^2-192*x^3)*d^2/dx^2 P(n-2,x) + (160+96*x)*P(n-1,x) + (38-32*x)*d/dx P(n-1,x) + (38*x-96*x^2)*d^2/dx^2 P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 3 + 20*x + 9*x^2, P(3,x) = 4 + 52*x + 81*x^2 + 16*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[2, x_] := 3 + 20*x + 9*x^2;\nP[3, x_] := 4 + 52*x + 81*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(152 + 876*x - 540*x^2 - 368*x^3 + 24*x^4)*P[n - 4, x] - (27 + 119*x - 45*x^2 - 432*x^3 - 128*x^4)*D[P[n - 4, x], x] - (27*x - 9*x^2 - 9*x^3 - 36*x^4)*D[P[n - 4, x], {x, 2}] + (568 + 1400*x - 1344*x^2 + 96*x^3)*P[n - 3, x] + (92 + 396*x - 650*x^2 - 1056*x^3)*D[P[n - 3, x], x] + (92*x - 76*x^2 - 130*x^3 - 96*x^4)*D[P[n - 3, x], {x, 2}] - (596 + 336*x + 144*x^2)*P[n - 2, x] - (103 + 300*x - 960*x^2)*D[P[n - 2, x], x] - (103*x - 152*x^2 - 192*x^3)*D[P[n - 2, x], {x, 2}] + (160 + 96*x)*P[n - 1, x] + (38 - 32*x)*D[P[n - 1, x], x] + (38*x - 96*x^2)*D[P[n - 1, x], {x, 2}])/(24)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; v[3] = 3 + 20*x + 9*x^2; v[4] = 4 + 52*x + 81*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(152 + 876*x - 540*x^2 - 368*x^3 + 24*x^4)*v[i - 4] - (27 + 119*x - 45*x^2 - 432*x^3 - 128*x^4)*deriv(v[i - 4], x) - (27*x - 9*x^2 - 9*x^3 - 36*x^4)*deriv(deriv(v[i - 4], x), x) + (568 + 1400*x - 1344*x^2 + 96*x^3)*v[i - 3] + (92 + 396*x - 650*x^2 - 1056*x^3)*deriv(v[i - 3], x) + (92*x - 76*x^2 - 130*x^3 - 96*x^4)*deriv(deriv(v[i - 3], x), x) - (596 + 336*x + 144*x^2)*v[i - 2] - (103 + 300*x - 960*x^2)*deriv(v[i - 2], x) - (103*x - 152*x^2 - 192*x^3)*deriv(deriv(v[i - 2], x), x) + (160 + 96*x)*v[i - 1] + (38 - 32*x)*deriv(v[i - 1], x) + (38*x - 96*x^2)*deriv(deriv(v[i - 1], x), x))/(24)); 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-9fa98aec39d32ffa","steps":["(\u03b8+1)\u00b2"],"edges":["e-68b833aed5ed70d7"],"text":"A104029 \u2192 (\u03b8+1)\u00b2"},"headline":"A104029 \u2192 (\u03b8+1)\u00b2","chain":null}