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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":"e0357b0e24939980378ef9bd6fc65102e1113fdacfb2ebfdd2a9c28cfeb4a9c6","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-ef0a5df5e949a196","role":"sequence","name":"Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*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,      4;\n  2: 1,     44,         9;\n  3: 1,    216,       486,         16;\n  4: 1,    916,      7101,       3664,          25;\n  5: 1,   3728,     71775,     127600,       23300,          36;\n  6: 1,  14988,    617652,    2589872,     1715700,      134892,          49;\n  7: 1,  60040,   4888296,   40556144,    63368975,    19553184,      735490,         64;\n  8: 1, 240260,  36885951,  547900448,  1659733375,  1232776008,   200823511,    3844160,       81;\n  9: 1, 961152, 270499797, 6751000848, 35464152000, 51068378880, 20674940097, 1923554112, 19463328, 100;\n","recurrence_sentence":"The row polynomials satisfy (36*n^2-84*n+49)*P(n,x) = (2916*n^2-3888*n+1701)*x^2*P(n-4,x) + (1296*n^2-1728*n+819)*x^3*d/dx P(n-4,x) - ((432*n^2-576*n-1428)*x-(5832*n^3-18144*n^2+15282*n-4095)*x^2)*P(n-3,x) + ((432*n^2-576*n+651)*x^2+(2592*n^3-12312*n^2+12555*n-4557)*x^3)*d/dx P(n-3,x) + ((1296*n^2-1728*n+819)*x^3-(1296*n^2-1728*n+819)*x^4)*d^2/dx^2 P(n-3,x) + (72*n^2-96*n+14-(432*n^3-1728*n^2+540*n+1792)*x+(2916*n^4-11340*n^3+14013*n^2-4914*n-735)*x^2)*P(n-2,x) - ((216*n^2-288*n-462)*x-(432*n^3-576*n-1904)*x^2-(1296*n^4-9288*n^3+15768*n^2-7098*n-833)*x^3)*d/dx P(n-2,x) - ((324*n^2-432*n-315)*x^2-(1296*n^3-2592*n^2+1080*n-168)*x^3+(1296*n^3-2916*n^2+1512*n+147)*x^4)*d^2/dx^2 P(n-2,x) - (36*n^2-12*n-35-(432*n^3-1296*n^2+1260*n-392)*x)*P(n-1,x) + ((216*n^2-396*n+42)*x-(432*n^3-864*n^2+252*n+196)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 44*x + 9*x^2, P(3,x) = 1 + 216*x + 486*x^2 + 16*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[2, x_] := 1 + 44*x + 9*x^2;\nP[3, x_] := 1 + 216*x + 486*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((2916*n^2 - 3888*n + 1701)*x^2*P[n - 4, x] + (1296*n^2 - 1728*n + 819)*x^3*D[P[n - 4, x], x] - ((432*n^2 - 576*n - 1428)*x - (5832*n^3 - 18144*n^2 + 15282*n - 4095)*x^2)*P[n - 3, x] + ((432*n^2 - 576*n + 651)*x^2 + (2592*n^3 - 12312*n^2 + 12555*n - 4557)*x^3)*D[P[n - 3, x], x] + ((1296*n^2 - 1728*n + 819)*x^3 - (1296*n^2 - 1728*n + 819)*x^4)*D[P[n - 3, x], {x, 2}] + (72*n^2 - 96*n + 14 - (432*n^3 - 1728*n^2 + 540*n + 1792)*x + (2916*n^4 - 11340*n^3 + 14013*n^2 - 4914*n - 735)*x^2)*P[n - 2, x] - ((216*n^2 - 288*n - 462)*x - (432*n^3 - 576*n - 1904)*x^2 - (1296*n^4 - 9288*n^3 + 15768*n^2 - 7098*n - 833)*x^3)*D[P[n - 2, x], x] - ((324*n^2 - 432*n - 315)*x^2 - (1296*n^3 - 2592*n^2 + 1080*n - 168)*x^3 + (1296*n^3 - 2916*n^2 + 1512*n + 147)*x^4)*D[P[n - 2, x], {x, 2}] - (36*n^2 - 12*n - 35 - (432*n^3 - 1296*n^2 + 1260*n - 392)*x)*P[n - 1, x] + ((216*n^2 - 396*n + 42)*x - (432*n^3 - 864*n^2 + 252*n + 196)*x^2)*D[P[n - 1, x], x])/(36*n^2 - 84*n + 49)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; v[3] = 1 + 44*x + 9*x^2; v[4] = 1 + 216*x + 486*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((2916*n^2 - 3888*n + 1701)*x^2*v[i - 4] + (1296*n^2 - 1728*n + 819)*x^3*deriv(v[i - 4], x) - ((432*n^2 - 576*n - 1428)*x - (5832*n^3 - 18144*n^2 + 15282*n - 4095)*x^2)*v[i - 3] + ((432*n^2 - 576*n + 651)*x^2 + (2592*n^3 - 12312*n^2 + 12555*n - 4557)*x^3)*deriv(v[i - 3], x) + ((1296*n^2 - 1728*n + 819)*x^3 - (1296*n^2 - 1728*n + 819)*x^4)*deriv(deriv(v[i - 3], x), x) + (72*n^2 - 96*n + 14 - (432*n^3 - 1728*n^2 + 540*n + 1792)*x + (2916*n^4 - 11340*n^3 + 14013*n^2 - 4914*n - 735)*x^2)*v[i - 2] - ((216*n^2 - 288*n - 462)*x - (432*n^3 - 576*n - 1904)*x^2 - (1296*n^4 - 9288*n^3 + 15768*n^2 - 7098*n - 833)*x^3)*deriv(v[i - 2], x) - ((324*n^2 - 432*n - 315)*x^2 - (1296*n^3 - 2592*n^2 + 1080*n - 168)*x^3 + (1296*n^3 - 2916*n^2 + 1512*n + 147)*x^4)*deriv(deriv(v[i - 2], x), x) - (36*n^2 - 12*n - 35 - (432*n^3 - 1296*n^2 + 1260*n - 392)*x)*v[i - 1] + ((216*n^2 - 396*n + 42)*x - (432*n^3 - 864*n^2 + 252*n + 196)*x^2)*deriv(v[i - 1], x))/(36*n^2 - 84*n + 49)); 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-ef0a5df5e949a196","steps":["(\u03b8+1)\u00b2"],"edges":["e-6dfcea353fc0a182"],"text":"A144440 \u2192 (\u03b8+1)\u00b2"},"headline":"A144440 \u2192 (\u03b8+1)\u00b2","chain":null}