A356812 Expansion of e.g.f. exp(x * (1 - exp(2*x))).

1, 0, -4, -12, 16, 400, 2208, -448, -131840, -1357056, -4820480, 71120896, 1537308672, 14006460416, 3075702784, -2224350781440, -41354996154368, -359660395495424, 1675436608585728, 121894823709900800, 2317859245604208640, 20543311167964053504

Offset: 0

Author

Seiichi Manyama, Aug 29 2022

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..499

Crossrefs

Cf. A292893, A356813.

Cf. A351736, A356815, A356819.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Exp[x(1-Exp[2x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 04 2023 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-exp(2*x)))))

PARI

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(2*k+1)*x)^(k+1)))

PARI

a(n) = sum(k=0, n, (-1)^k*(2*k+1)^(n-k)*binomial(n, k));

PARI

a(n) = n!*sum(k=0, n\2, (-1)^k*2^(n-k)*stirling(n-k, k, 2)/(n-k)!);

Formula

G.f.: Sum_{k>=0} (-x)^k / (1 - (2*k+1)*x)^(k+1).

a(n) = Sum_{k=0..n} (-1)^k * (2*k+1)^(n-k) * binomial(n,k).

a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * 2^(n-k) * Stirling2(n-k,k)/(n-k)!.