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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":"57d6410573c28340dcee1066caaf4651ed71f35e75c35fc0edc4e6ecf329cbcc","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,         24;\n  2:       12,        204,         486;\n  3:       60,       1596,        7938,        10368;\n  4:      360,      13176,      101493,       273888,        243000;\n  5:     2520,     118152,     1244079,      5164992,       9404100,       6298560;\n  6:    20160,    1156896,    15535314,     87765552,     244709100,     334768464,      180033840;\n  7:   181440,   12347424,   202290210,   1452730032,    5493590100,   11470062672,    12556074384,     5643509760;\n  8:  1814400,  143069760,  2773008108,  24237230400,  116222949225,  328291409808,   547085547036,   499228904448,   192849310080;\n  9: 19958400, 1791495360, 40160297988, 414503300160, 2414572616475, 8632036914408, 19423173584556, 26928339383808, 21076439876640, 7142567040000;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(12*n^3+24*n^2+12*n)*x-(81*n^3+81*n^2)*x^2)*P(n-2,x) + ((12*n^3+24*n^2+12*n)*x^2+(36*n^3+36*n^2)*x^3)*d/dx P(n-2,x) + (2*n^2+4*n+1+(12*n^2+12*n)*x)*P(n-1,x) - (12*n^2+12*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 24*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 24*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (12*n^3 + 24*n^2 + 12*n)*x - (81*n^3 + 81*n^2)*x^2)*P[n - 2, x] + ((12*n^3 + 24*n^2 + 12*n)*x^2 + (36*n^3 + 36*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^2 + 4*n + 1 + (12*n^2 + 12*n)*x)*P[n - 1, x] - (12*n^2 + 12*n)*x^2*D[P[n - 1, x], 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 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (12*n^3 + 24*n^2 + 12*n)*x - (81*n^3 + 81*n^2)*x^2)*v[i - 2] + ((12*n^3 + 24*n^2 + 12*n)*x^2 + (36*n^3 + 36*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^2 + 4*n + 1 + (12*n^2 + 12*n)*x)*v[i - 1] - (12*n^2 + 12*n)*x^2*deriv(v[i - 1], x))/(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)\u00b2"],"edges":["e-45b4054949a52789","e-7e4d3471dea1d9fe"],"text":"A145324 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A145324 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}