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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":"7af88bd40ff424037a8e6108cbc20b8a8cac9a63bd5c50fc1634af94a9f4c523","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f425461cbf5769b3","role":"sequence","name":"Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     3,        16;\n   3:     9,       160,         288;\n   4:    27,      1264,        6048,         6144;\n   5:    81,      9280,       89280,       221184,        153600;\n   6:   243,     66256,     1149120,      5222400,       8448000,        4423680;\n   7:   729,    467680,    13832928,    102850560,     291072000,      345047040,      144506880;\n   8:  2187,   3285424,   160580448,   1837860864,    8101632000,    16318955520,    15173222400,     5284823040;\n   9:  6561,  23032960,  1825522560,  30993629184,  199877529600,   608662978560,   942762885120,   718735933440,   214035333120;\n  10: 19683, 161335696, 20495341440, 503848919040, 4572513792000, 19755721359360, 45337588531200, 56758999449600, 36600041963520, 9512681472000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+16*x-32*x^2)*P(n-2,x) + (4*x-12*x^2-16*x^3)*d/dx P(n-2,x) + (2+16*x)*P(n-1,x) + (4*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 16*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 16*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 16*x - 32*x^2)*P[n - 2, x] + (4*x - 12*x^2 - 16*x^3)*D[P[n - 2, x], x] + (2 + 16*x)*P[n - 1, x] + (4*x + 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 16*x; for(n = 3, N, my(i = n); v[i] = (3 + 16*x - 32*x^2)*v[i - 2] + (4*x - 12*x^2 - 16*x^3)*deriv(v[i - 2], x) + (2 + 16*x)*v[i - 1] + (4*x + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f425461cbf5769b3","steps":["(\u03b8+1)\u00b2"],"edges":["e-102428afa3452ef2"],"text":"A225473 \u2192 (\u03b8+1)\u00b2"},"headline":"A225473 \u2192 (\u03b8+1)\u00b2","chain":null}