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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":"014fffdbeb102e08552d05915ab4857a56293c56470f8b144559bbe1221907d6","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-61f790c1436a0d7c","role":"sequence","name":"Triangle T(n,k) of associated Stirling numbers of second kind, n >= 2, 1 <= k <= floor(n/2).","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;\n  2: 1,    6;\n  3: 1,   20;\n  4: 1,   50,    45;\n  5: 1,  112,   315;\n  6: 1,  238,  1470,    420;\n  7: 1,  492,  5754,   5040;\n  8: 1, 1002, 20475,  37800,  4725;\n  9: 1, 2024, 68805, 227920, 86625;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-1)*x^2*P(n-4,x) - (n+1)*x^2*d/dx P(n-3,x) + (2*n+2)*x*P(n-2,x) + P(n-1,x) + x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 6*x, P(3,x) = 1 + 20*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 6*x;\nP[3, x_] := 1 + 20*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 1)*x^2*P[n - 4, x] - (n + 1)*x^2*D[P[n - 3, x], x] + (2*n + 2)*x*P[n - 2, x] + P[n - 1, x] + x*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] = 1; v[3] = 1 + 6*x; v[4] = 1 + 20*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 1)*x^2*v[i - 4] - (n + 1)*x^2*deriv(v[i - 3], x) + (2*n + 2)*x*v[i - 2] + v[i - 1] + x*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-61f790c1436a0d7c","steps":["\u03b8+1"],"edges":["e-f21edc6325e9777b"],"text":"A008299 \u2192 \u03b8+1"},"headline":"A008299 \u2192 \u03b8+1","chain":null}