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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":"7e4a0bec7654e6017b0dd3f6d3dfa315c12ce55edf88970c2d1268cfa87cbb63","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-33a50ee3b762dcdc","role":"sequence","name":"basis_map_A008288 of Triangle of coefficients in expansion of (1+2*x)^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:     3,      4;\n  2:     9,     32,      12;\n  3:    27,    164,     156,      32;\n  4:    81,    704,    1176,     576,      80;\n  5:   243,   2756,    6840,    5760,    1840,     192;\n  6:   729,  10208,   34212,   42688,   22960,    5376,     448;\n  7:  2187,  36452,  155220,  262944,  207440,   80448,   14784,   1024;\n  8:  6561, 126848,  658224, 1428864, 1524960,  858624,  258048,  38912,  2304;\n  9: 19683, 433028, 2657136, 7095808, 9707040, 7350912, 3179008, 776192, 99072, 5120;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9-12*x+4*x^2)*P(n-4,x) + (24-4*x-8*x^2)*P(n-3,x) - (22+12*x+4*x^2)*P(n-2,x) + (8+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 9 + 32*x + 12*x^2, P(3,x) = 27 + 164*x + 156*x^2 + 32*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[2, x_] := 9 + 32*x + 12*x^2;\nP[3, x_] := 27 + 164*x + 156*x^2 + 32*x^3;\nP[n_, x_] := P[n, x] = Expand[-(9 - 12*x + 4*x^2)*P[n - 4, x] + (24 - 4*x - 8*x^2)*P[n - 3, x] - (22 + 12*x + 4*x^2)*P[n - 2, x] + (8 + 4*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] = 3 + 4*x; v[3] = 9 + 32*x + 12*x^2; v[4] = 27 + 164*x + 156*x^2 + 32*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(9 - 12*x + 4*x^2)*v[i - 4] + (24 - 4*x - 8*x^2)*v[i - 3] - (22 + 12*x + 4*x^2)*v[i - 2] + (8 + 4*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-71965e79e47a4526","steps":["basis A008288","\u03b8+1"],"edges":["e-9c90c90150c15b0e","e-099086ec53021eaa"],"text":"A013609 \u2192 basis A008288 \u2192 \u03b8+1"},"headline":"A013609 \u2192 basis A008288 \u2192 \u03b8+1","chain":null}