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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":"22bc0c287f4815fa5834c74d3995c71f46b7feac99ae955beccd753f4e2ba553","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-70594f2875298da0","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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:        3,        12;\n  2:       12,       102,         162;\n  3:       60,       798,        2646,         2592;\n  4:      360,      6588,       33831,        68472,        48600;\n  5:     2520,     59076,      414693,      1291248,      1880820,       1049760;\n  6:    20160,    578448,     5178438,     21941388,     48941820,      55794744,      25719120;\n  7:   181440,   6173712,    67430070,    363182508,   1098718020,    1911677112,    1793724912,     705438720;\n  8:  1814400,  71534880,   924336036,   6059307600,  23244589845,   54715234968,   78155078148,   62403613056,   21427701120;\n  9: 19958400, 895747680, 13386765996, 103625825040, 482914523295, 1438672819068, 2774739083508, 3366042422976, 2341826652960, 714256704000;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(6*n^3+12*n^2+6*n)*x+(9*n^3+9*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(6*n^2+6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (6*n^3 + 12*n^2 + 6*n)*x + (9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (6*n^2 + 6*n)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (6*n^3 + 12*n^2 + 6*n)*x + (9*n^3 + 9*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (6*n^2 + 6*n)*x)*v[i - 1])/(n)); 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-2feb6dc01f424b0a","steps":["basis A013610","\u03b8+1"],"edges":["e-45b4054949a52789","e-7c7c33c3985c0844"],"text":"A145324 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A145324 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}