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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":"6ede5548c3ce72de06cbfc4d701597d998e0e7112bb5ac89577322db2de31615","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7e652d8a57b6dcff","role":"sequence","name":"Triangle of 4-Eulerian numbers.","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: 1,    16;\n  2: 1,    52,     144;\n  3: 1,   128,    1017,     1024;\n  4: 1,   284,    4779,    13136,      6400;\n  5: 1,   600,   18810,   103520,    134625,      36864;\n  6: 1,  1236,   67230,   648160,   1643625,    1190484,     200704;\n  7: 1,  2512,  226719,  3548800,  15307375,   21343824,    9506833,    1048576;\n  8: 1,  5068,  736677, 17822704, 120896875,  282361284,  240855727,   70474048,   5308416;\n  9: 1, 10184, 2335464, 84387296, 855116150, 3086899704, 4376265656, 2451314816, 493744329, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (n+3)*P(n,x) = -((4*n^2+24*n+32)*x-(9*n^3+79*n^2+224*n+204)*x^2)*P(n-2,x) - ((2*n+8)*x-(4*n^2+32*n+65)*x^2-(4*n^3+22*n^2+11*n-57)*x^3)*d/dx P(n-2,x) - ((n+4)*x^2-(4*n^2+28*n+49)*x^3+(4*n^2+27*n+45)*x^4)*d^2/dx^2 P(n-2,x) + (n+3+(4*n^2+24*n+36)*x)*P(n-1,x) + ((2*n+7)*x-(4*n^2+28*n+48)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 + 24*n + 32)*x - (9*n^3 + 79*n^2 + 224*n + 204)*x^2)*P[n - 2, x] - ((2*n + 8)*x - (4*n^2 + 32*n + 65)*x^2 - (4*n^3 + 22*n^2 + 11*n - 57)*x^3)*D[P[n - 2, x], x] - ((n + 4)*x^2 - (4*n^2 + 28*n + 49)*x^3 + (4*n^2 + 27*n + 45)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 3 + (4*n^2 + 24*n + 36)*x)*P[n - 1, x] + ((2*n + 7)*x - (4*n^2 + 28*n + 48)*x^2)*D[P[n - 1, x], x])/(n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-((4*n^2 + 24*n + 32)*x - (9*n^3 + 79*n^2 + 224*n + 204)*x^2)*v[i - 2] - ((2*n + 8)*x - (4*n^2 + 32*n + 65)*x^2 - (4*n^3 + 22*n^2 + 11*n - 57)*x^3)*deriv(v[i - 2], x) - ((n + 4)*x^2 - (4*n^2 + 28*n + 49)*x^3 + (4*n^2 + 27*n + 45)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 3 + (4*n^2 + 24*n + 36)*x)*v[i - 1] + ((2*n + 7)*x - (4*n^2 + 28*n + 48)*x^2)*deriv(v[i - 1], x))/(n + 3)); 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-7e652d8a57b6dcff","steps":["(\u03b8+1)\u00b2"],"edges":["e-eb4a65a7d3859b93"],"text":"A144698 \u2192 (\u03b8+1)\u00b2"},"headline":"A144698 \u2192 (\u03b8+1)\u00b2","chain":null}