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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":"7ece880d3712f1071f2c3bf4590659a3b7fda4ea31ea674a83216adc745f1010","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a2304b34e998691c","role":"sequence","name":"Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,   8;\n   3: 1,  20,   30;\n   4: 1,  36,  135,    80;\n   5: 1,  56,  378,   560,    175;\n   6: 1,  80,  840,  2240,   1750,    336;\n   7: 1, 108, 1620,  6720,   9450,   4536,    588;\n   8: 1, 140, 2835, 16800,  36750,  31752,  10290,    960;\n   9: 1, 176, 4620, 36960, 115500, 155232,  90552,  21120,  1485;\n  10: 1, 216, 7128, 73920, 311850, 598752, 543312, 228096, 40095, 2200;\n","recurrence_sentence":"The row polynomials satisfy -(n^2+n-2-(n^4+2*n^3-3*n^2-4*n+4)*x)*P(n,x) = -(n^2+n-2-(n^4+2*n^3-9*n^2-6*n)*x-(3*n^4+8*n^3+9*n-2)*x^2)*P(n-1,x) - ((2*n+1)*x-(2*n^3+6*n^2+2*n+2)*x^2+(2*n^3+6*n^2+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + n - 2 - (n^4 + 2*n^3 - 9*n^2 - 6*n)*x - (3*n^4 + 8*n^3 + 9*n - 2)*x^2)*P[n - 1, x] - ((2*n + 1)*x - (2*n^3 + 6*n^2 + 2*n + 2)*x^2 + (2*n^3 + 6*n^2 + 1)*x^3)*D[P[n - 1, x], x])/(-n^2 - n + 2 + (n^4 + 2*n^3 - 3*n^2 - 4*n + 4)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n^2 + n - 2 - (n^4 + 2*n^3 - 9*n^2 - 6*n)*x - (3*n^4 + 8*n^3 + 9*n - 2)*x^2)*v[i - 1] - ((2*n + 1)*x - (2*n^3 + 6*n^2 + 2*n + 2)*x^2 + (2*n^3 + 6*n^2 + 1)*x^3)*deriv(v[i - 1], x))/(-n^2 - n + 2 + (n^4 + 2*n^3 - 3*n^2 - 4*n + 4)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a2304b34e998691c","steps":["\u03b8+1"],"edges":["e-78be3a4437894f86"],"text":"A062145 \u2192 \u03b8+1"},"headline":"A062145 \u2192 \u03b8+1","chain":null}