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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":"ec0ffe65dc6151e77875997766f433f65bb7d912681fd29917a23becd3b26da6","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-2f0c73de4c291a40","role":"sequence","name":"basis_map_A013611 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,         16;\n  2:       12,        136,         288;\n  3:       60,       1064,        4704,         6144;\n  4:      360,       8784,       60144,       162304,        153600;\n  5:     2520,      78768,      737232,      3060736,       5944320,       4423680;\n  6:    20160,     771264,     9206112,     52009216,     154680320,     235118592,      144506880;\n  7:   181440,    8231616,   119875680,    860877056,    3472491520,    8055791616,    10078322688,     5284823040;\n  8:  1814400,   95379840,  1643264064,  14362803200,   73464382720,  230569549824,   439126474752,   467499220992,   214035333120;\n  9: 19958400, 1194330240, 23798695104, 245631585280, 1526248369920, 6062555418624, 15590303547392, 25216844562432, 23391853608960, 9512681472000;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(8*n^3+16*n^2+8*n)*x+(16*n^3+16*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(8*n^2+8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (8*n^3 + 16*n^2 + 8*n)*x + (16*n^3 + 16*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (8*n^2 + 8*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 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (8*n^3 + 16*n^2 + 8*n)*x + (16*n^3 + 16*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (8*n^2 + 8*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 A013611","\u03b8+1"],"edges":["e-39cad9e518a532bb","e-9c6bcbb703cf55b9"],"text":"A145324 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A145324 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}