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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":"9e03b736b813e5785baa0c7ae1bca5ef3cf7c13f00e6b839a32be76dc0f569c5","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-f5630447d9b46bac","role":"sequence","name":"basis_map_A008288 of Triangle of coefficients in expansion of (2+3x)^n.","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:       5,        6;\n  2:      25,       78,        27;\n  3:     125,      666,       567,       108;\n  4:     625,     4782,      7047,      3132,       405;\n  5:    3125,    31314,     68175,     51300,     14985,      1458;\n  6:   15625,   193878,    569187,    628668,    304965,     65610,      5103;\n  7:   78125,  1156506,   4315923,   6437124,   4534785,   1596510,    270459,    17496;\n  8:  390625,  6719262,  30596319,  58313628,  55190565,  27942570,   7659603,  1067256,   59049;\n  9: 1953125, 38282274, 206455959, 483473124, 583801425, 394644150, 154299411, 34484616, 4074381, 196830;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(100-60*x+9*x^2)*P(n-4,x) + (140+18*x-18*x^2)*P(n-3,x) - (69+36*x+9*x^2)*P(n-2,x) + (14+6*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 5 + 6*x, P(2,x) = 25 + 78*x + 27*x^2, P(3,x) = 125 + 666*x + 567*x^2 + 108*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 6*x;\nP[2, x_] := 25 + 78*x + 27*x^2;\nP[3, x_] := 125 + 666*x + 567*x^2 + 108*x^3;\nP[n_, x_] := P[n, x] = Expand[-(100 - 60*x + 9*x^2)*P[n - 4, x] + (140 + 18*x - 18*x^2)*P[n - 3, x] - (69 + 36*x + 9*x^2)*P[n - 2, x] + (14 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 6*x; v[3] = 25 + 78*x + 27*x^2; v[4] = 125 + 666*x + 567*x^2 + 108*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(100 - 60*x + 9*x^2)*v[i - 4] + (140 + 18*x - 18*x^2)*v[i - 3] - (69 + 36*x + 9*x^2)*v[i - 2] + (14 + 6*x)*v[i - 1]); 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-5733d79f45a14084","steps":["basis A008288","\u03b8+1"],"edges":["e-28556ceb40718bbf","e-4934132220cc6f5d"],"text":"A013620 \u2192 basis A008288 \u2192 \u03b8+1"},"headline":"A013620 \u2192 basis A008288 \u2192 \u03b8+1","chain":null}