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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":"126336c50131cb02143913336bbd3e14226b2285dedeb88311bbffe572795d4e","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-723ec2078be12613","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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,       2;\n  2: 1,      28,         3;\n  3: 1,     166,       249,           4;\n  4: 1,     864,      4650,        1728,           5;\n  5: 1,    4362,     59694,       79592,       10905,           6;\n  6: 1,   21860,    653661,     2094192,     1089435,       65580,           7;\n  7: 1,  109358,   6597711,    42359180,    52948975,    13195422,      382753,          8;\n  8: 1,  546856,  63611484,   738891440,  1810736110,  1108337160,   148426796,    2187424,        9;\n  9: 1, 2734354, 596947740, 11761159312, 50390210070, 60468252084, 20582028796, 1591860640, 12304593, 10;\n","recurrence_sentence":"The row polynomials satisfy (12*n-13)*P(n,x) = -(192*n-16)*x^2*P(n-4,x) + ((288*n-24)*x-(384*n^2-704*n-8)*x^2)*P(n-3,x) - ((192*n-16)*x^2-(192*n-16)*x^3)*d/dx P(n-3,x) + (36*n-3+(288*n^2-432*n-62)*x-(192*n^3-496*n^2+228*n+91)*x^2)*P(n-2,x) + ((144*n-12)*x-(192*n^2-16*n-64)*x^2+(192*n^2-160*n-52)*x^3)*d/dx P(n-2,x) - (24*n+10-(96*n^2-176*n+78)*x)*P(n-1,x) + ((48*n-52)*x-(48*n-52)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 28*x + 3*x^2, P(3,x) = 1 + 166*x + 249*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 1 + 28*x + 3*x^2;\nP[3, x_] := 1 + 166*x + 249*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(192*n - 16)*x^2*P[n - 4, x] + ((288*n - 24)*x - (384*n^2 - 704*n - 8)*x^2)*P[n - 3, x] - ((192*n - 16)*x^2 - (192*n - 16)*x^3)*D[P[n - 3, x], x] + (36*n - 3 + (288*n^2 - 432*n - 62)*x - (192*n^3 - 496*n^2 + 228*n + 91)*x^2)*P[n - 2, x] + ((144*n - 12)*x - (192*n^2 - 16*n - 64)*x^2 + (192*n^2 - 160*n - 52)*x^3)*D[P[n - 2, x], x] - (24*n + 10 - (96*n^2 - 176*n + 78)*x)*P[n - 1, x] + ((48*n - 52)*x - (48*n - 52)*x^2)*D[P[n - 1, x], x])/(12*n - 13)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 28*x + 3*x^2; v[4] = 1 + 166*x + 249*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(192*n - 16)*x^2*v[i - 4] + ((288*n - 24)*x - (384*n^2 - 704*n - 8)*x^2)*v[i - 3] - ((192*n - 16)*x^2 - (192*n - 16)*x^3)*deriv(v[i - 3], x) + (36*n - 3 + (288*n^2 - 432*n - 62)*x - (192*n^3 - 496*n^2 + 228*n + 91)*x^2)*v[i - 2] + ((144*n - 12)*x - (192*n^2 - 16*n - 64)*x^2 + (192*n^2 - 160*n - 52)*x^3)*deriv(v[i - 2], x) - (24*n + 10 - (96*n^2 - 176*n + 78)*x)*v[i - 1] + ((48*n - 52)*x - (48*n - 52)*x^2)*deriv(v[i - 1], x))/(12*n - 13)); 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-723ec2078be12613","steps":["\u03b8+1"],"edges":["e-d85c389038a64bf9"],"text":"A144441 \u2192 \u03b8+1"},"headline":"A144441 \u2192 \u03b8+1","chain":null}