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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":"441b70975238bd9e8818166b50c0808e51a054ae61d97dc4d0bdfb73def9436f","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-6aadffa116a44288","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/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,      12;\n  2:     12,      84,       81;\n  3:     48,     480,      999,       432;\n  4:    192,    2496,     7992,      8208,      2025;\n  5:    768,   12288,    53136,     88128,     54675,      8748;\n  6:   3072,   58368,   317952,    732672,    759780,    317844,     35721;\n  7:  12288,  270336,  1776384,   5253120,   7737120,   5552064,   1678887,    139968;\n  8:  49152, 1228800,  9455616,  34172928,  65577600,  68024448,  36006768,   8258112,   531441;\n  9: 196608, 5505024, 48549888, 207470592, 491961600, 673339392, 522710496, 213311232, 38440899, 1968300;\n","recurrence_sentence":"The row polynomials satisfy -(4*n^2-6*n+2+(6*n^3-11*n^2-2*n+3)*x-(6*n^3+3*n^2)*x^2)*P(n,x) = -(16*n^2-24*n+8+(24*n^3+28*n^2-108*n+44)*x+(30*n^3-69*n^2-90*n+54)*x^2-(54*n^3+81*n^2-18*n-9)*x^3)*P(n-1,x) + ((8*n-4)*x+(12*n^2+32*n-22)*x^2+(24*n^2-36)*x^3-(36*n^2+72*n+18)*x^4)*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[(-(16*n^2 - 24*n + 8 + (24*n^3 + 28*n^2 - 108*n + 44)*x + (30*n^3 - 69*n^2 - 90*n + 54)*x^2 - (54*n^3 + 81*n^2 - 18*n - 9)*x^3)*P[n - 1, x] + ((8*n - 4)*x + (12*n^2 + 32*n - 22)*x^2 + (24*n^2 - 36)*x^3 - (36*n^2 + 72*n + 18)*x^4)*D[P[n - 1, x], x])/(-4*n^2 + 6*n - 2 - (6*n^3 - 11*n^2 - 2*n + 3)*x + (6*n^3 + 3*n^2)*x^2)]];\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] = (-(16*n^2 - 24*n + 8 + (24*n^3 + 28*n^2 - 108*n + 44)*x + (30*n^3 - 69*n^2 - 90*n + 54)*x^2 - (54*n^3 + 81*n^2 - 18*n - 9)*x^3)*v[i - 1] + ((8*n - 4)*x + (12*n^2 + 32*n - 22)*x^2 + (24*n^2 - 36)*x^3 - (36*n^2 + 72*n + 18)*x^4)*deriv(v[i - 1], x))/(-4*n^2 + 6*n - 2 - (6*n^3 - 11*n^2 - 2*n + 3)*x + (6*n^3 + 3*n^2)*x^2)); 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-c85abc56078671cc","steps":["basis A013610","\u03b8+1"],"edges":["e-12f73d6f0ea24cf5","e-f03065b8c5c63876"],"text":"A232535 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A232535 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}