A371009 Expansion of e.g.f. (1/x) * Series_Reversion( 2*x/(3 - exp(2*x)) ).

1, -1, 0, 8, -16, -336, 2464, 30176, -572160, -3654400, 193852928, -29664768, -88869165056, 788014352384, 51013392617472, -1125131950034944, -33201578814668800, 1536045242886979584, 19518336239699623936, -2267097378027280924672

Offset: 0

Author

Seiichi Manyama, Mar 08 2024

Keywords

• sign

Links

• Index entries for reversions of series

Crossrefs

Cf. A371007, A371008.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(2*x/(3-exp(2*x)))/x))

PARI

a(n) = sum(k=0, n+1, (-1)^k*3^(n+1-k)*k^n*binomial(n+1, k))/(2*(n+1));

Formula

a(n) = 1/(2*(n+1)) * Sum_{k=0..n+1} (-1)^k * 3^(n+1-k) * k^n * binomial(n+1,k).

a(n) = n! * Sum_{k=0..n} (-1)^k * 2^(n-k) * Stirling2(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2024