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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":"d67532c11213acb99044e2ae89d8aa7c4df1c31f06da37e7f6f8f5e142a5f999","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-7e99bca717f629d0","role":"sequence","name":"veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","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:      0;\n  1:      2;\n  2:     12;\n  3:     54,     16;\n  4:    216,    192;\n  5:    810,   1440,     96;\n  6:   2916,   8640,   1728;\n  7:  10206,  45360,  18144,    512;\n  8:  34992, 217728, 145152,  12288;\n  9: 118098, 979776, 979776, 165888, 2560;\n","recurrence_sentence":"The row polynomials satisfy -(27*n-27-(6*n^3-12*n^2+6*n)*x)*P(n,x) = -(81*n-(18*n^3-18*n^2-72*n+144)*x-(8*n^3-8*n^2-16*n)*x^2)*P(n-1,x) - (162*x-(36*n^2+36*n+72)*x^2+(16*n^2+16*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(81*n - (18*n^3 - 18*n^2 - 72*n + 144)*x - (8*n^3 - 8*n^2 - 16*n)*x^2)*P[n - 1, x] - (162*x - (36*n^2 + 36*n + 72)*x^2 + (16*n^2 + 16*n)*x^3)*D[P[n - 1, x], x])/(-27*n + 27 + (6*n^3 - 12*n^2 + 6*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(81*n - (18*n^3 - 18*n^2 - 72*n + 144)*x - (8*n^3 - 8*n^2 - 16*n)*x^2)*v[i - 1] - (162*x - (36*n^2 + 36*n + 72)*x^2 + (16*n^2 + 16*n)*x^3)*deriv(v[i - 1], x))/(-27*n + 27 + (6*n^3 - 12*n^2 + 6*n)*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-01cea33b6485d9cc","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-252e19fd44a7da1c","e-169e53acef46d16b"],"text":"A038220 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A038220 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}