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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. 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":"80ad41b718c5274440f0d7f2719091ba2df93ee0158d9bb962b26f1776bfaf9c","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-ff9680a2cfd23e93","role":"sequence","name":"Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.","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,  2;\n  1: 1,  6;\n  2: 1, 10,   6;\n  3: 1, 14,  24;\n  4: 1, 18,  54,   16;\n  5: 1, 22,  96,   80;\n  6: 1, 26, 150,  224,   40;\n  7: 1, 30, 216,  480,  240;\n  8: 1, 34, 294,  880,  800,  96;\n  9: 1, 38, 384, 1456, 2000, 672;\n","recurrence_sentence":"The row polynomials satisfy -(n^2+3*n+2+(2*n^3+8*n^2+4*n-8)*x-(2*n^3+18*n^2+52*n+48)*x^2)*P(n,x) = -(n^2+3*n+2+(2*n^3+16*n^2+22*n-4)*x+(6*n^3+20*n^2-26*n-64)*x^2-(8*n^3+64*n^2+152*n+96)*x^3)*P(n-1,x) + ((n+2)*x+(2*n^2+18*n+26)*x^2+(14*n^2+66*n+56)*x^3-(16*n^2+112*n+192)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 3*n + 2 + (2*n^3 + 16*n^2 + 22*n - 4)*x + (6*n^3 + 20*n^2 - 26*n - 64)*x^2 - (8*n^3 + 64*n^2 + 152*n + 96)*x^3)*P[n - 1, x] + ((n + 2)*x + (2*n^2 + 18*n + 26)*x^2 + (14*n^2 + 66*n + 56)*x^3 - (16*n^2 + 112*n + 192)*x^4)*D[P[n - 1, x], x])/(-n^2 - 3*n - 2 - (2*n^3 + 8*n^2 + 4*n - 8)*x + (2*n^3 + 18*n^2 + 52*n + 48)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 3*n + 2 + (2*n^3 + 16*n^2 + 22*n - 4)*x + (6*n^3 + 20*n^2 - 26*n - 64)*x^2 - (8*n^3 + 64*n^2 + 152*n + 96)*x^3)*v[i - 1] + ((n + 2)*x + (2*n^2 + 18*n + 26)*x^2 + (14*n^2 + 66*n + 56)*x^3 - (16*n^2 + 112*n + 192)*x^4)*deriv(v[i - 1], x))/(-n^2 - 3*n - 2 - (2*n^3 + 8*n^2 + 4*n - 8)*x + (2*n^3 + 18*n^2 + 52*n + 48)*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-ff9680a2cfd23e93","steps":["\u03b8+1"],"edges":["e-447bddf7f850df84"],"text":"A199478 \u2192 \u03b8+1"},"headline":"A199478 \u2192 \u03b8+1","chain":null}