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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":"5b0f066a1683a2b8427e2a9aa0767695a5fde8c83d333345d181118eac274a96","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-1f922d5ae3a4fc22","role":"sequence","name":"veronese_section_2_1 of Riordan array (1-x,x/(1-x)).","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: 0,  2;\n  2: 0,  4;\n  3: 0,  6,   3;\n  4: 0,  8,  12;\n  5: 0, 10,  30,   4;\n  6: 0, 12,  60,  24;\n  7: 0, 14, 105,  84,   5;\n  8: 0, 16, 168, 224,  40;\n  9: 0, 18, 252, 504, 180, 6;\n","recurrence_sentence":"The row polynomials satisfy -(12*n-12-(n^3+2*n^2-n-2)*x)*P(n,x) = -(12*n-24-(n^3+n^2-26*n)*x-(n^3+5*n^2+6*n)*x^2)*P(n-1,x) - (24*x-(2*n^2+10*n+36)*x^2+(2*n^2+10*n+12)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 24 - (n^3 + n^2 - 26*n)*x - (n^3 + 5*n^2 + 6*n)*x^2)*P[n - 1, x] - (24*x - (2*n^2 + 10*n + 36)*x^2 + (2*n^2 + 10*n + 12)*x^3)*D[P[n - 1, x], x])/(-12*n + 12 + (n^3 + 2*n^2 - n - 2)*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*x; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n - 24 - (n^3 + n^2 - 26*n)*x - (n^3 + 5*n^2 + 6*n)*x^2)*v[i - 1] - (24*x - (2*n^2 + 10*n + 36)*x^2 + (2*n^2 + 10*n + 12)*x^3)*deriv(v[i - 1], x))/(-12*n + 12 + (n^3 + 2*n^2 - n - 2)*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-7717563017cb9e5f","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-6fdf1329e5e61ed8","e-e7734fa965b454e2"],"text":"A159854 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A159854 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}