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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":"d8b57a75d97913593f9d10e63a889015ea802062d3dc0edd7ca4c909bc6222ec","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-4f94f81718ae3273","role":"sequence","name":"Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.","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:          13,           18,            3;\n   2:          75,          158,           54,            4;\n   3:         541,         1530,          795,          120,           5;\n   4:        4683,        16622,        11610,         2660,         225,           6;\n   5:       47293,       201978,       177303,        54600,        7000,         378,          7;\n   6:      545835,      2724878,      2881494,      1115604,      192150,       15708,        588,         8;\n   7:     7087261,     40492890,     50101635,     23378040,     5062155,      555660,      31458,       864,       9;\n   8:   102247563,    657944942,    932458770,    509327620,   132937875,    18341442,    1393560,     57840,    1215,     10;\n   9:  1622632573,  11611834938,  18546943503,  11609221800,  3547799750,   590007726,   56331429,   3136320,   99495,   1650,   11;\n  10: 28091567595, 221291822798, 393306236334, 277569511604, 97228909800, 18933361656, 2178661716, 152385816, 6482025, 162250, 2178, 12;\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) - 2*x^2*d^2/dx^2 P(n-2,x) + (3+2*x)*P(n-1,x) + (2+3*x)*d/dx P(n-1,x) + 2*x*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 13 + 18*x + 3*x^2, P(2,x) = 75 + 158*x + 54*x^2 + 4*x^3.","mathematica":"P[1, x_] := 13 + 18*x + 3*x^2;\nP[2, x_] := 75 + 158*x + 54*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 3*x^2*D[P[n - 2, x], x] - 2*x^2*D[P[n - 2, x], {x, 2}] + (3 + 2*x)*P[n - 1, x] + (2 + 3*x)*D[P[n - 1, x], x] + 2*x*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] = 13 + 18*x + 3*x^2; v[2] = 75 + 158*x + 54*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = -x^2*v[i - 2] - 3*x^2*deriv(v[i - 2], x) - 2*x^2*deriv(deriv(v[i - 2], x), x) + (3 + 2*x)*v[i - 1] + (2 + 3*x)*deriv(v[i - 1], x) + 2*x*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-4f94f81718ae3273","steps":["\u03b8+1"],"edges":["e-b0816fbf60f9dc02"],"text":"A088729 \u2192 \u03b8+1"},"headline":"A088729 \u2192 \u03b8+1","chain":null}