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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":"b4765e76a8750b749b4ef2bf846cb5479c9129b8adb0da6367d5ebf3e3133e6b","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-66d5ecabca128038","role":"sequence","name":"Triangle read by rows, (n-1) zeros followed by (2n, 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: 2,  4;\n  2: 0, 16,  9;\n  3: 0,  0, 54,  16;\n  4: 0,  0,  0, 128,  25;\n  5: 0,  0,  0,   0, 250,  36;\n  6: 0,  0,  0,   0,   0, 432,  49;\n  7: 0,  0,  0,   0,   0,   0, 686,   64;\n  8: 0,  0,  0,   0,   0,   0,   0, 1024,   81;\n  9: 0,  0,  0,   0,   0,   0,   0,    0, 1458, 100;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-6*n^2+6*n-2+n^2*x)*P(n,x) = (2*n^3*x+(n^2+2*n+1)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3*x + (n^2 + 2*n + 1)*x^2)*P[n - 1, x])/(2*n^3 - 6*n^2 + 6*n - 2 + n^2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^3*x + (n^2 + 2*n + 1)*x^2)*v[i - 1])/(2*n^3 - 6*n^2 + 6*n - 2 + n^2*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-6fea5aff4f44fa54","steps":["2D+1","(\u03b8+1)\u00b2"],"edges":["e-224384a428b81acf","e-6514f9242ec5adea"],"text":"A023531 \u2192 2D+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A023531 \u2192 2D+1 \u2192 (\u03b8+1)\u00b2","chain":null}