A341106 a(n) = 2^n*E2poly(n, -1/2), where E2poly(n, x) = Sum_{k=0..n} A340556(n, k)*x^k, are the second-order Eulerian polynomials.

1, -1, 0, 6, -12, -144, 1080, 5184, -127008, 95904, 19077120, -154929024, -3210337152, 70284900096, 391453171200, -30354545511936, 153830450875392, 13189520200402944, -244127117929789440, -5109022268709986304, 237988748560571301888, 571783124036801765376

Offset: 0

Author

Peter Luschny, Feb 13 2021

Keywords

• sign

Crossrefs

Cf. A340556.

Programs

Maple

E2poly := (n, x) -> add(A340556(n, k)*x^k, k = 0..n):
seq(2^n*E2poly(n, -1/2), n = 0..21);
# By series reversion:
serrev := proc(gf, len) series(gf, y, len);
gfun:-seriestoseries(%, 'revogf'); gfun:-seriestolist(%);
gfun:-listtolist(%, 'Laplace'); subsop(1 = NULL, %) end:
gf := (6*y + exp(3*y) - 1)/9: serrev(gf, 23);

Mathematica

R := 22; f[y_] := (6y + Exp[3y] - 1)/9;
S := InverseSeries[Series[f[y], {y, 0, R}], x];
Drop[CoefficientList[S, x] Table[n!, {n, 0, R}], 1]