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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":"7349946486b07139b3a92a9f047ae6e06b9a96c39df67777ea3f9ed3c01d5107","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e9164c616dcc1bbd","role":"sequence","name":"Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.","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:      1,        2;\n   3:      2,       12,         3;\n   4:      6,       76,        45,         4;\n   5:     24,      544,       564,       112,         5;\n   6:    120,     4400,      7020,      2320,       225,        6;\n   7:    720,    39840,     90840,     44160,      6950,      396,       7;\n   8:   5040,   399840,   1240680,    827680,    187250,    17052,     637,     8;\n   9:  40320,  4408320,  17969280,  15724800,   4779600,   616896,   36456,   960,    9;\n  10: 362880, 52980480, 276151680, 307005440, 120027600, 20541024, 1704024, 70464, 1377, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-7*n+10+(2*n-10)*x+x^2)*P(n-2,x) - ((3*n-15)*x+3*x^2)*d/dx P(n-2,x) + (2*n-6+2*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 10 + (2*n - 10)*x + x^2)*P[n - 2, x] - ((3*n - 15)*x + 3*x^2)*D[P[n - 2, x], x] + (2*n - 6 + 2*x)*P[n - 1, x] + 3*x*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] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = -(n^2 - 7*n + 10 + (2*n - 10)*x + x^2)*v[i - 2] - ((3*n - 15)*x + 3*x^2)*deriv(v[i - 2], x) + (2*n - 6 + 2*x)*v[i - 1] + 3*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-e9164c616dcc1bbd","steps":["\u03b8+1"],"edges":["e-b6e15b128f476a59"],"text":"A322944 \u2192 \u03b8+1"},"headline":"A322944 \u2192 \u03b8+1","chain":null}