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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":"8ff4b89060e4d21d92f7d07d2a99b5c685eeba1b8f12759bc7efed3bf15cc2a8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-522def1e8ebf49cd","role":"sequence","name":"Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the 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: 1;\n  1: 1,  2;\n  2: 1,  8,   3;\n  3: 1, 14,  27,    4;\n  4: 1, 20,  78,   64,     5;\n  5: 1, 26, 156,  280,   125,     6;\n  6: 1, 32, 261,  760,   775,   216,     7;\n  7: 1, 38, 393, 1612,  2765,  1806,   343,    8;\n  8: 1, 44, 552, 2944,  7310,  8232,  3724,  512,   9;\n  9: 1, 50, 738, 4864, 16030, 26676, 21168, 7008, 729, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n+(3*n^3-6*n^2-3*n+6)*x-(20*n^3-9*n^2+7*n-12)*x^2+(32*n^3-40*n^2+12*n)*x^3)*P(n,x) = (n^2+n+(3*n^3+6*n^2-n-4)*x-(2*n^3+27*n^2+n-24)*x^2-(52*n^3+35*n^2+9*n-88)*x^3+(96*n^3-8*n^2-116*n+36)*x^4)*P(n-1,x) - ((n+1)*x+(3*n^2+11*n+11)*x^2+(13*n^2+12*n+23)*x^3-(80*n^2+88*n+11)*x^4+(64*n^2+64*n-24)*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[((n^2 + n + (3*n^3 + 6*n^2 - n - 4)*x - (2*n^3 + 27*n^2 + n - 24)*x^2 - (52*n^3 + 35*n^2 + 9*n - 88)*x^3 + (96*n^3 - 8*n^2 - 116*n + 36)*x^4)*P[n - 1, x] - ((n + 1)*x + (3*n^2 + 11*n + 11)*x^2 + (13*n^2 + 12*n + 23)*x^3 - (80*n^2 + 88*n + 11)*x^4 + (64*n^2 + 64*n - 24)*x^5)*D[P[n - 1, x], x])/(n^2 + n + (3*n^3 - 6*n^2 - 3*n + 6)*x - (20*n^3 - 9*n^2 + 7*n - 12)*x^2 + (32*n^3 - 40*n^2 + 12*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] = ((n^2 + n + (3*n^3 + 6*n^2 - n - 4)*x - (2*n^3 + 27*n^2 + n - 24)*x^2 - (52*n^3 + 35*n^2 + 9*n - 88)*x^3 + (96*n^3 - 8*n^2 - 116*n + 36)*x^4)*v[i - 1] - ((n + 1)*x + (3*n^2 + 11*n + 11)*x^2 + (13*n^2 + 12*n + 23)*x^3 - (80*n^2 + 88*n + 11)*x^4 + (64*n^2 + 64*n - 24)*x^5)*deriv(v[i - 1], x))/(n^2 + n + (3*n^3 - 6*n^2 - 3*n + 6)*x - (20*n^3 - 9*n^2 + 7*n - 12)*x^2 + (32*n^3 - 40*n^2 + 12*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-522def1e8ebf49cd","steps":["\u03b8+1"],"edges":["e-b53062d4155caae9"],"text":"A209414 \u2192 \u03b8+1"},"headline":"A209414 \u2192 \u03b8+1","chain":null}