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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.
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%I #10 Dec 10 2023 17:26:43

%S 1,-1,0,6,-12,-144,1080,5184,-127008,95904,19077120,-154929024,

%T -3210337152,70284900096,391453171200,-30354545511936,153830450875392,

%U 13189520200402944,-244127117929789440,-5109022268709986304,237988748560571301888,571783124036801765376

%N 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.

%p E2poly := (n, x) -> add(A340556(n, k)*x^k, k = 0..n):

%p seq(2^n*E2poly(n, -1/2), n = 0..21);

%p # By series reversion:

%p serrev := proc(gf, len) series(gf, y, len);

%p gfun:-seriestoseries(%, 'revogf'); gfun:-seriestolist(%);

%p gfun:-listtolist(%, 'Laplace'); subsop(1 = NULL, %) end:

%p gf := (6*y + exp(3*y) - 1)/9: serrev(gf, 23);

%t R := 22; f[y_] := (6y + Exp[3y] - 1)/9;

%t S := InverseSeries[Series[f[y], {y, 0, R}], x];

%t Drop[CoefficientList[S, x] Table[n!, {n, 0, R}], 1]

%Y Cf. A340556.

%K sign

%O 0,4

%A _Peter Luschny_, Feb 13 2021