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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":"c25ac518b3522bfaafdfd7a9895e0d2cc116907fc159cbcac9742389de807176","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-b9dfb234567937e9","role":"sequence","name":"basis_map_A007318 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*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:     3,      2;\n  2:    10,     14,      3;\n  3:    33,     70,     33,      4;\n  4:   109,    310,    228,     60,      5;\n  5:   360,   1284,   1287,    532,     95,      6;\n  6:  1189,   5100,   6480,   3676,   1030,    138,     7;\n  7:  3927,  19682,  30303,  21916,   8445,   1770,   189,    8;\n  8: 12970,  74378, 134562, 118468,  58355,  16818,  2800,  248,   9;\n  9: 42837, 276610, 575079, 597140, 358990, 132384, 30275, 4168, 315, 10;\n","recurrence_sentence":"The row polynomials satisfy -(26*n+(13*n^2-21*n)*x+(3*n^3-17*n^2-36*n)*x^2-(3*n^3+8*n^2+5*n)*x^3)*P(n,x) = -(208*n-130+(143*n^2+209*n-324)*x+(24*n^3+122*n^2-44*n-268)*x^2+(12*n^3+5*n^2-81*n-84)*x^3-(6*n^2+16*n+10)*x^4)*P(n-1,x) + (169+(169*n+533)*x+(39*n^2+403*n+622)*x^2+(69*n^2+327*n+326)*x^3+(33*n^2+101*n+73)*x^4+(3*n^2+8*n+5)*x^5)*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[(-(208*n - 130 + (143*n^2 + 209*n - 324)*x + (24*n^3 + 122*n^2 - 44*n - 268)*x^2 + (12*n^3 + 5*n^2 - 81*n - 84)*x^3 - (6*n^2 + 16*n + 10)*x^4)*P[n - 1, x] + (169 + (169*n + 533)*x + (39*n^2 + 403*n + 622)*x^2 + (69*n^2 + 327*n + 326)*x^3 + (33*n^2 + 101*n + 73)*x^4 + (3*n^2 + 8*n + 5)*x^5)*D[P[n - 1, x], x])/(-26*n - (13*n^2 - 21*n)*x - (3*n^3 - 17*n^2 - 36*n)*x^2 + (3*n^3 + 8*n^2 + 5*n)*x^3)]];\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] = (-(208*n - 130 + (143*n^2 + 209*n - 324)*x + (24*n^3 + 122*n^2 - 44*n - 268)*x^2 + (12*n^3 + 5*n^2 - 81*n - 84)*x^3 - (6*n^2 + 16*n + 10)*x^4)*v[i - 1] + (169 + (169*n + 533)*x + (39*n^2 + 403*n + 622)*x^2 + (69*n^2 + 327*n + 326)*x^3 + (33*n^2 + 101*n + 73)*x^4 + (3*n^2 + 8*n + 5)*x^5)*deriv(v[i - 1], x))/(-26*n - (13*n^2 - 21*n)*x - (3*n^3 - 17*n^2 - 36*n)*x^2 + (3*n^3 + 8*n^2 + 5*n)*x^3)); 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-d5c14b9fec875ecc","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-03e3acc3469b904b","e-f99d818d47615a6e"],"text":"A121574 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A121574 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}