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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":"57111e359b0168c156f4f645328bd68d7501abf2fcdb9aa59fe3c90205f16c50","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-df88941ef8650983","role":"sequence","name":"one_plus_derivative of Stirling2 triangle with scaled diagonals (powers of 2).","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,       2;\n  2:     10,      16,        3;\n  3:     36,     104,       45,        4;\n  4:    136,     640,      480,       96,        5;\n  5:    528,    3872,     4500,     1520,      175,       6;\n  6:   2080,   23296,    39648,    20160,     3850,     288,      7;\n  7:   8256,  139904,   337680,   243264,    68600,    8400,    441,     8;\n  8:  32896,  839680,  2818560,  2774016,  1085280,  193536,  16464,   640,   9;\n  9: 131328, 5038592, 23227200, 30503680, 15934800, 3884832, 476280, 29760, 891, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(32-4*x^2-4*x^3+x^4)*P(n-4,x) - (16*x-8*x^3+2*x^4)*d/dx P(n-4,x) + (56-6*x^2)*P(n-3,x) + (24*x-4*x^2-2*x^3)*d/dx P(n-3,x) - (36-2*x^2)*P(n-2,x) - (12*x-2*x^2)*d/dx P(n-2,x) + 10*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 10 + 16*x + 3*x^2, P(3,x) = 36 + 104*x + 45*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[2, x_] := 10 + 16*x + 3*x^2;\nP[3, x_] := 36 + 104*x + 45*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-(32 - 4*x^2 - 4*x^3 + x^4)*P[n - 4, x] - (16*x - 8*x^3 + 2*x^4)*D[P[n - 4, x], x] + (56 - 6*x^2)*P[n - 3, x] + (24*x - 4*x^2 - 2*x^3)*D[P[n - 3, x], x] - (36 - 2*x^2)*P[n - 2, x] - (12*x - 2*x^2)*D[P[n - 2, x], x] + 10*P[n - 1, x] + 2*x*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 + 2*x; v[3] = 10 + 16*x + 3*x^2; v[4] = 36 + 104*x + 45*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(32 - 4*x^2 - 4*x^3 + x^4)*v[i - 4] - (16*x - 8*x^3 + 2*x^4)*deriv(v[i - 4], x) + (56 - 6*x^2)*v[i - 3] + (24*x - 4*x^2 - 2*x^3)*deriv(v[i - 3], x) - (36 - 2*x^2)*v[i - 2] - (12*x - 2*x^2)*deriv(v[i - 2], x) + 10*v[i - 1] + 2*x*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-a7e9ebb9860a9dbe","steps":["D+1","\u03b8+1"],"edges":["e-9fe96d700c9448ac","e-1450b3b9f407a14f"],"text":"A075497 \u2192 D+1 \u2192 \u03b8+1"},"headline":"A075497 \u2192 D+1 \u2192 \u03b8+1","chain":null}