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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":"fea6a2ff5c81b59bacdaa8b76a2f5fb48852dd91e6ee694defa41bc0be3b7833","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-387e148a64b95986","role":"sequence","name":"Row reversed version of triangle A201637 (second-order Eulerian triangle).","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: 0,       4;\n  2: 0,       8,        9;\n  3: 0,      24,       72,        16;\n  4: 0,      96,      522,       352,        25;\n  5: 0,     480,     3996,      5248,      1300,        36;\n  6: 0,    2880,    33372,     70400,     36300,      4104,       49;\n  7: 0,   20160,   305856,    930240,    803000,    201960,    11760,      64;\n  8: 0,  161280,  3070224,  12564864,  16100500,   7048800,   977550,   31616,    81;\n  9: 0, 1451520, 33605280, 176420736, 311001600, 207558000, 52062500, 4304640, 81324, 100;\n","recurrence_sentence":"The row polynomials satisfy n^2*P(n,x) = -(n^4-4*n^3+n^2+6*n+(4*n^4-8*n^3-8*n+12)*x-(9*n^4-2*n^3-7*n^2)*x^2)*P(n-2,x) - ((2*n^3-n^2-8*n+3)*x-(4*n^4-4*n^3-4*n^2+10*n-15)*x^2-(4*n^4-14*n^3-13*n^2)*x^3)*d/dx P(n-2,x) - ((n^2+2*n-3)*x^2-(4*n^3+4*n^2+2*n-3)*x^3+(4*n^3+3*n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3-3*n^2-3*n+4*n^3*x)*P(n-1,x) + ((2*n^2+2*n-3)*x-(4*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - 4*n^3 + n^2 + 6*n + (4*n^4 - 8*n^3 - 8*n + 12)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*P[n - 2, x] - ((2*n^3 - n^2 - 8*n + 3)*x - (4*n^4 - 4*n^3 - 4*n^2 + 10*n - 15)*x^2 - (4*n^4 - 14*n^3 - 13*n^2)*x^3)*D[P[n - 2, x], x] - ((n^2 + 2*n - 3)*x^2 - (4*n^3 + 4*n^2 + 2*n - 3)*x^3 + (4*n^3 + 3*n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 - 3*n^2 - 3*n + 4*n^3*x)*P[n - 1, x] + ((2*n^2 + 2*n - 3)*x - (4*n^3 + 4*n^2)*x^2)*D[P[n - 1, x], x])/(n^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 - 4*n^3 + n^2 + 6*n + (4*n^4 - 8*n^3 - 8*n + 12)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*v[i - 2] - ((2*n^3 - n^2 - 8*n + 3)*x - (4*n^4 - 4*n^3 - 4*n^2 + 10*n - 15)*x^2 - (4*n^4 - 14*n^3 - 13*n^2)*x^3)*deriv(v[i - 2], x) - ((n^2 + 2*n - 3)*x^2 - (4*n^3 + 4*n^2 + 2*n - 3)*x^3 + (4*n^3 + 3*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 - 3*n^2 - 3*n + 4*n^3*x)*v[i - 1] + ((2*n^2 + 2*n - 3)*x - (4*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^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-e2f0f447a502f269","steps":["reverse (shift +1)","(\u03b8+1)\u00b2"],"edges":["e-5f52c7cc32f7742d","e-2f1c4f1a3b1c7046"],"text":"A201637 \u2192 reverse (shift +1) \u2192 (\u03b8+1)\u00b2"},"headline":"A201637 \u2192 reverse (shift +1) \u2192 (\u03b8+1)\u00b2","chain":null}