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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":"d1ff4b5227f9fc8457f74cd3e831d0fe0677984f665c26f783b9b93ca54f922b","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-b2f14d9f44a39df2","role":"sequence","name":"Triangle of coefficients of Legendre polynomials 2^n P_n (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;\n  2:     6,      4;\n  3:    20,     24;\n  4:    70,    120,     18;\n  5:   252,    560,    180;\n  6:   924,   2520,   1260,    80;\n  7:  3432,  11088,   7560,  1120;\n  8: 12870,  48048,  41580, 10080,  350;\n  9: 48620, 205920, 216216, 73920, 6300;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+2*n+(n^3+n^2)*x)*P(n,x) = (16*n^2-4+(4*n^3+18*n^2-10*n-8)*x+(2*n^3+4*n^2-2*n-4)*x^2)*P(n-1,x) - ((16*n+8)*x+(4*n^2+28*n+16)*x^2+(4*n^2+12*n+8)*x^3)*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[((16*n^2 - 4 + (4*n^3 + 18*n^2 - 10*n - 8)*x + (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*P[n - 1, x] - ((16*n + 8)*x + (4*n^2 + 28*n + 16)*x^2 + (4*n^2 + 12*n + 8)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (n^3 + n^2)*x)]];\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] = ((16*n^2 - 4 + (4*n^3 + 18*n^2 - 10*n - 8)*x + (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*v[i - 1] - ((16*n + 8)*x + (4*n^2 + 28*n + 16)*x^2 + (4*n^2 + 12*n + 8)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (n^3 + n^2)*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-b2f14d9f44a39df2","steps":["\u03b8+1"],"edges":["e-64544209412c0a1a"],"text":"A008556 \u2192 \u03b8+1"},"headline":"A008556 \u2192 \u03b8+1","chain":null}