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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":"3590f398ff71a5359b7a6ce9517aa6c83888403d58bc5e89253cd3f7a150e33c","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-91d76a88748fc9ba","role":"sequence","name":"Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.","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:     14,       28,        6;\n  1:     42,      144,       81,         8;\n  2:    132,      660,      660,       176,        10;\n  3:    429,     2860,     4290,      2080,       325,        12;\n  4:   1430,    12012,    24570,     18200,      5250,       540,       14;\n  5:   4862,    49504,   129948,    133280,     59500,     11424,      833,       16;\n  6:  16796,   201552,   651168,    868224,    542640,    162792,    22344,     1216,      18;\n  7:  58786,   813960,  3139560,   5209344,   4273290,   1809864,   391020,    40320,    1701,     20;\n  8: 208012,  3268760, 14709420,  29418840,  30284100,  16959096,  5206740,   850080,   68310,   2300,   22;\n  9: 742900, 13075040, 67418175, 158631000, 198288750, 140263200, 57274140, 13358400, 1707750, 110000, 3025, 24;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+26*n+42+(4*n^3+34*n^2+94*n+84)*x)*P(n,x) = (14*n^2+80*n+116+(15*n^3+135*n^2+390*n+360)*x+(12*n^3+94*n^2+238*n+196)*x^2)*P(n-1,x) + (n+2+(n^2+10*n+22)*x+(7*n^2+37*n+46)*x^2-(8*n^2+48*n+70)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 28*x + 6*x^2.","mathematica":"P[0, x_] := 14 + 28*x + 6*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((14*n^2 + 80*n + 116 + (15*n^3 + 135*n^2 + 390*n + 360)*x + (12*n^3 + 94*n^2 + 238*n + 196)*x^2)*P[n - 1, x] + (n + 2 + (n^2 + 10*n + 22)*x + (7*n^2 + 37*n + 46)*x^2 - (8*n^2 + 48*n + 70)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 26*n + 42 + (4*n^3 + 34*n^2 + 94*n + 84)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 14 + 28*x + 6*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n^2 + 80*n + 116 + (15*n^3 + 135*n^2 + 390*n + 360)*x + (12*n^3 + 94*n^2 + 238*n + 196)*x^2)*v[i - 1] + (n + 2 + (n^2 + 10*n + 22)*x + (7*n^2 + 37*n + 46)*x^2 - (8*n^2 + 48*n + 70)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 26*n + 42 + (4*n^3 + 34*n^2 + 94*n + 84)*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-91d76a88748fc9ba","steps":["\u03b8+1"],"edges":["e-4e5d17bd5b20b967"],"text":"A286785 \u2192 \u03b8+1"},"headline":"A286785 \u2192 \u03b8+1","chain":null}