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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":"0904ae0fbedeb7165a3b1821e99ae7a53206a3f8eb70d6cf635b20aa139492c4","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-9401e1a9db1eb1a5","role":"sequence","name":"Triangle of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.","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:   2,       2;\n   3:   4,      16,         3;\n   4:   8,     104,        60,         4;\n   5:  16,     640,       876,       160,         5;\n   6:  32,    3872,     11472,      4368,       350,         6;\n   7:  64,   23296,    143472,    102656,     15960,       672,       7;\n   8: 128,  139904,   1756608,   2244416,    607120,     47376,    1176,      8;\n   9: 256,  839680,  21289152,  47230464,  21019120,   2718336,  121128,   1920,    9;\n  10: 512, 5038592, 256729344, 972994816, 689611680, 139393056, 9954560, 276672, 2970, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 3*x^2*d/dx P(n-2,x) - x^3*d^2/dx^2 P(n-2,x) + (2+2*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 3*x^2*D[P[n - 2, x], x] - x^3*D[P[n - 2, x], {x, 2}] + (2 + 2*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = -x^2*v[i - 2] - 3*x^2*deriv(v[i - 2], x) - x^3*deriv(deriv(v[i - 2], x), x) + (2 + 2*x)*v[i - 1] + 4*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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-9401e1a9db1eb1a5","steps":["\u03b8+1"],"edges":["e-fe4fc1e0efd4be7c"],"text":"A071951 \u2192 \u03b8+1"},"headline":"A071951 \u2192 \u03b8+1","chain":null}