A370910 Expansion of e.g.f. (1/x) * Series_Reversion( 4*x/(1 + 3*exp(4*x)) ).

1, 3, 30, 534, 13944, 483528, 20979696, 1095048048, 66870491520, 4679735191680, 369376085620992, 32469964691563776, 3146279097351097344, 333211862918094763008, 38295176235674130905088, 4746999846951047891417088, 631337582065248312276910080

Offset: 0

Author

Seiichi Manyama, Mar 05 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A201595, A370909.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(4*x/(1+3*exp(4*x)))/x))

PARI

a(n) = sum(k=0, n+1, 3^k*k^n*binomial(n+1, k))/(4*(n+1));

Formula

a(n) = 1/(4*(n+1)) * Sum_{k=0..n+1} 3^k * k^n * binomial(n+1,k).

a(n) = n! * Sum_{k=0..n} 3^k * 4^(n-k) * Stirling2(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2024