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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":"8b5e17f2e49039e472f6a30a4d6981f4f78e3b31b0de9bb4086a2dabed2affdc","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-d75669a8cd7f9393","role":"sequence","name":"A133080 * A007318.","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: 2,  4;\n  2: 1,  8,   9;\n  3: 2, 20,  36,   16;\n  4: 1, 16,  54,   64,   25;\n  5: 2, 36, 144,  224,  150,   36;\n  6: 1, 24, 135,  320,  375,  216,   49;\n  7: 2, 52, 324,  880, 1250,  972,  392,   64;\n  8: 1, 32, 252,  896, 1750, 2016, 1372,  512,  81;\n  9: 2, 68, 576, 2240, 4900, 6552, 5488, 2816, 810, 100;\n","recurrence_sentence":"The row polynomials satisfy (4+(10*n-4)*x+(14*n^2-18*n+4)*x^2+(6*n^3-8*n^2)*x^3+(n^4-2*n^3+n^2)*x^4)*P(n,x) = -(12*n-20+(51*n^2-93*n+14)*x+(60*n^3-110*n^2+22*n+4)*x^2+(24*n^4-18*n^3-48*n^2+18*n+28)*x^3+(3*n^5+9*n^4-25*n^3-3*n^2+16*n+8)*x^4+(2*n^5-3*n^4-2*n^3+4*n^2+n-2)*x^5)*P(n-1,x) + (3+(18*n-4)*x+(33*n^2+n-11)*x^2+(18*n^3+40*n^2-30*n-12)*x^3+(3*n^4+28*n^3+2*n^2-19*n-9)*x^4+(5*n^4+10*n^3-7*n^2-6*n)*x^5+(2*n^4-2*n^2+1)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 20 + (51*n^2 - 93*n + 14)*x + (60*n^3 - 110*n^2 + 22*n + 4)*x^2 + (24*n^4 - 18*n^3 - 48*n^2 + 18*n + 28)*x^3 + (3*n^5 + 9*n^4 - 25*n^3 - 3*n^2 + 16*n + 8)*x^4 + (2*n^5 - 3*n^4 - 2*n^3 + 4*n^2 + n - 2)*x^5)*P[n - 1, x] + (3 + (18*n - 4)*x + (33*n^2 + n - 11)*x^2 + (18*n^3 + 40*n^2 - 30*n - 12)*x^3 + (3*n^4 + 28*n^3 + 2*n^2 - 19*n - 9)*x^4 + (5*n^4 + 10*n^3 - 7*n^2 - 6*n)*x^5 + (2*n^4 - 2*n^2 + 1)*x^6)*D[P[n - 1, x], x])/(4 + (10*n - 4)*x + (14*n^2 - 18*n + 4)*x^2 + (6*n^3 - 8*n^2)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(12*n - 20 + (51*n^2 - 93*n + 14)*x + (60*n^3 - 110*n^2 + 22*n + 4)*x^2 + (24*n^4 - 18*n^3 - 48*n^2 + 18*n + 28)*x^3 + (3*n^5 + 9*n^4 - 25*n^3 - 3*n^2 + 16*n + 8)*x^4 + (2*n^5 - 3*n^4 - 2*n^3 + 4*n^2 + n - 2)*x^5)*v[i - 1] + (3 + (18*n - 4)*x + (33*n^2 + n - 11)*x^2 + (18*n^3 + 40*n^2 - 30*n - 12)*x^3 + (3*n^4 + 28*n^3 + 2*n^2 - 19*n - 9)*x^4 + (5*n^4 + 10*n^3 - 7*n^2 - 6*n)*x^5 + (2*n^4 - 2*n^2 + 1)*x^6)*deriv(v[i - 1], x))/(4 + (10*n - 4)*x + (14*n^2 - 18*n + 4)*x^2 + (6*n^3 - 8*n^2)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)); 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-d75669a8cd7f9393","steps":["(\u03b8+1)\u00b2"],"edges":["e-7c04308011ed96cd"],"text":"A133087 \u2192 (\u03b8+1)\u00b2"},"headline":"A133087 \u2192 (\u03b8+1)\u00b2","chain":null}