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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":"d426f9bc98ff867bdea659734d6c1c1e4301794dda57a6a582262b48cd33ec6d","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,      8;\n  2:     9,     64,      36;\n  3:    27,    328,     468,      128;\n  4:    81,   1408,    3528,     2304,      400;\n  5:   243,   5512,   20520,    23040,     9200,     1152;\n  6:   729,  20416,  102636,   170752,   114800,    32256,     3136;\n  7:  2187,  72904,  465660,  1051776,  1037200,   482688,   103488,    8192;\n  8:  6561, 253696, 1974672,  5715456,  7624800,  5151744,  1806336,  311296,  20736;\n  9: 19683, 866056, 7971408, 28383232, 48535200, 44105472, 22253056, 6209536, 891648, 51200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9-24*x-36*x^2)*P(n-4,x) - (24*x^2-16*x^3)*d/dx P(n-4,x) + (24-8*x+72*x^2)*P(n-3,x) + (8*x^2+32*x^3)*d/dx P(n-3,x) - (22+24*x-36*x^2)*P(n-2,x) + (24*x^2+16*x^3)*d/dx P(n-2,x) + (8+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 8*x, P(2,x) = 9 + 64*x + 36*x^2, P(3,x) = 27 + 328*x + 468*x^2 + 128*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 8*x;\nP[2, x_] := 9 + 64*x + 36*x^2;\nP[3, x_] := 27 + 328*x + 468*x^2 + 128*x^3;\nP[n_, x_] := P[n, x] = Expand[-(9 - 24*x - 36*x^2)*P[n - 4, x] - (24*x^2 - 16*x^3)*D[P[n - 4, x], x] + (24 - 8*x + 72*x^2)*P[n - 3, x] + (8*x^2 + 32*x^3)*D[P[n - 3, x], x] - (22 + 24*x - 36*x^2)*P[n - 2, x] + (24*x^2 + 16*x^3)*D[P[n - 2, x], x] + (8 + 8*x)*P[n - 1, x] - 8*x^2*D[P[n - 1, x], 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 + 8*x; v[3] = 9 + 64*x + 36*x^2; v[4] = 27 + 328*x + 468*x^2 + 128*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(9 - 24*x - 36*x^2)*v[i - 4] - (24*x^2 - 16*x^3)*deriv(v[i - 4], x) + (24 - 8*x + 72*x^2)*v[i - 3] + (8*x^2 + 32*x^3)*deriv(v[i - 3], x) - (22 + 24*x - 36*x^2)*v[i - 2] + (24*x^2 + 16*x^3)*deriv(v[i - 2], x) + (8 + 8*x)*v[i - 1] - 8*x^2*deriv(v[i - 1], 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-71965e79e47a4526","steps":["basis A008288","(\u03b8+1)\u00b2"],"edges":["e-9c90c90150c15b0e","e-24e0a4701d67fa69"],"text":"A013609 \u2192 basis A008288 \u2192 (\u03b8+1)\u00b2"},"headline":"A013609 \u2192 basis A008288 \u2192 (\u03b8+1)\u00b2","chain":null}