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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":"0fb59b02fcdc8677a0ff910fa797f2a8d34f257d9dd56a0bc626809441050ee4","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-e41c30a965e7cb9c","role":"sequence","name":"Exponential Riordan array [1, log((1-x)/(1-2x))].","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: 0,        4;\n  2: 0,       12,         9;\n  3: 0,       56,        81,        16;\n  4: 0,      360,       747,       288,        25;\n  5: 0,     2976,      7830,      4400,       750,       36;\n  6: 0,    30240,     94266,     68400,     17125,     1620,      49;\n  7: 0,   365760,   1295028,   1134784,    370125,    51660,    3087,     64;\n  8: 0,  5140800,  20100204,  20387136,   8107225,  1481760,  131026,   5376,   81;\n  9: 0, 82414080, 348928560, 397929152, 184974300, 41450724, 4815720, 292992, 8748, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^4-48*n^3+212*n^2-408*n+288)*P(n-4,x) + (12*n^3-90*n^2+222*n-180+(8*n^2-40*n+48)*x)*P(n-3,x) - (8*n^2-40*n+48)*x^2*d/dx P(n-3,x) - (13*n^2-52*n+52+(12*n-24)*x-9*x^2)*P(n-2,x) + ((12*n-24)*x^2+4*x^3)*d/dx P(n-2,x) + (6*n-9+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4*x, P(2,x) = 12*x + 9*x^2, P(3,x) = 56*x + 81*x^2 + 16*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x;\nP[2, x_] := 12*x + 9*x^2;\nP[3, x_] := 56*x + 81*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[-(4*n^4 - 48*n^3 + 212*n^2 - 408*n + 288)*P[n - 4, x] + (12*n^3 - 90*n^2 + 222*n - 180 + (8*n^2 - 40*n + 48)*x)*P[n - 3, x] - (8*n^2 - 40*n + 48)*x^2*D[P[n - 3, x], x] - (13*n^2 - 52*n + 52 + (12*n - 24)*x - 9*x^2)*P[n - 2, x] + ((12*n - 24)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (6*n - 9 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; v[3] = 12*x + 9*x^2; v[4] = 56*x + 81*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(4*n^4 - 48*n^3 + 212*n^2 - 408*n + 288)*v[i - 4] + (12*n^3 - 90*n^2 + 222*n - 180 + (8*n^2 - 40*n + 48)*x)*v[i - 3] - (8*n^2 - 40*n + 48)*x^2*deriv(v[i - 3], x) - (13*n^2 - 52*n + 52 + (12*n - 24)*x - 9*x^2)*v[i - 2] + ((12*n - 24)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (6*n - 9 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], 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-e41c30a965e7cb9c","steps":["(\u03b8+1)\u00b2"],"edges":["e-ddbda95705f62f87"],"text":"A131222 \u2192 (\u03b8+1)\u00b2"},"headline":"A131222 \u2192 (\u03b8+1)\u00b2","chain":null}