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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":"12749193c3abeca4f20a8139f7c0798c472b919cf9bb30cbbf7bc00bbee17e28","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-fefc87d068bda822","role":"sequence","name":"Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.","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,               4,               9;\n  3: 0,              32,              81,              16;\n  4: 0,             864,            2259,             576,             25;\n  5: 0,           55296,          146520,           40880,           2500,           36;\n  6: 0,         6912000,        18439416,         5370480,         376375,         8100,         49;\n  7: 0,      1492992000,      3998465856,      1192804864,       89688375,      2291580,      21609,       64;\n  8: 0,    512096256000,   1374833020608,    416240452096,    32626870225,    915163200,   10530982,    50176,     81;\n  9: 0, 262193283072000, 705066723127296, 215559259065344, 17355333261600, 515546251524, 6637501584, 39444864, 104976, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^6-12*n^5+60*n^4-160*n^3+240*n^2-192*n+64+(4*n^3-24*n^2+48*n-32)*x-9*x^2)*P(n-2,x) + ((4*n^3-24*n^2+48*n-32)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n^3-9*n^2+15*n-9+4*x)*P(n-1,x) - 4*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[-(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (4*n^3 - 24*n^2 + 48*n - 32)*x - 9*x^2)*P[n - 2, x] + ((4*n^3 - 24*n^2 + 48*n - 32)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n^3 - 9*n^2 + 15*n - 9 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];\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^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (4*n^3 - 24*n^2 + 48*n - 32)*x - 9*x^2)*v[i - 2] + ((4*n^3 - 24*n^2 + 48*n - 32)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n^3 - 9*n^2 + 15*n - 9 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], x)); 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-fefc87d068bda822","steps":["(\u03b8+1)\u00b2"],"edges":["e-28ef45b19d5a9989"],"text":"A269947 \u2192 (\u03b8+1)\u00b2"},"headline":"A269947 \u2192 (\u03b8+1)\u00b2","chain":null}