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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":"7afe9698e1270d9096ffa6c3763e1efe3bf4f5698c612bbf592c2da0c251569c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ee90405a2d071e0a","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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:    2,     6;\n  3:    5,    30,     27;\n  4:   13,   114,    216,    108;\n  5:   34,   390,   1134,   1188,     405;\n  6:   89,  1260,   4941,   7992,    5670,    1458;\n  7:  233,  3924,  19359,  42552,   46575,   24786,    5103;\n  8:  610, 11910,  70794, 197100,  293625,  240570,  102060,  17496;\n  9: 1597, 35466, 246618, 831924, 1573425, 1753974, 1143072, 402408, 59049;\n","recurrence_sentence":"The row polynomials satisfy -(60*n-90+(144*n-216)*x-(54*n^3-243*n^2+351*n-162)*x^2)*P(n,x) = -(90*n-120+(45*n^2+261*n-378)*x-(81*n^3-351*n^2+54*n+216)*x^2-(81*n^3-243*n^2+162*n)*x^3)*P(n-1,x) - (25+210*x-(45*n^2-90*n-477)*x^2-(162*n^2-324*n-216)*x^3-(81*n^2-162*n)*x^4)*d/dx 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[(-(90*n - 120 + (45*n^2 + 261*n - 378)*x - (81*n^3 - 351*n^2 + 54*n + 216)*x^2 - (81*n^3 - 243*n^2 + 162*n)*x^3)*P[n - 1, x] - (25 + 210*x - (45*n^2 - 90*n - 477)*x^2 - (162*n^2 - 324*n - 216)*x^3 - (81*n^2 - 162*n)*x^4)*D[P[n - 1, x], x])/(-60*n + 90 - (144*n - 216)*x + (54*n^3 - 243*n^2 + 351*n - 162)*x^2)]];\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] = (-(90*n - 120 + (45*n^2 + 261*n - 378)*x - (81*n^3 - 351*n^2 + 54*n + 216)*x^2 - (81*n^3 - 243*n^2 + 162*n)*x^3)*v[i - 1] - (25 + 210*x - (45*n^2 - 90*n - 477)*x^2 - (162*n^2 - 324*n - 216)*x^3 - (81*n^2 - 162*n)*x^4)*deriv(v[i - 1], x))/(-60*n + 90 - (144*n - 216)*x + (54*n^3 - 243*n^2 + 351*n - 162)*x^2)); 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-0eba401b117f16a9","steps":["basis A013610","\u03b8+1"],"edges":["e-0b8f676aef1665e7","e-155f06cdbb1d6dde"],"text":"A085478 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A085478 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}