A386769 Expansion of (1/x) * Series_Reversion( x * (1-2*x) / (1+3*x)^2 ).

1, 8, 89, 1162, 16646, 253218, 4016769, 65713602, 1100773166, 18786755128, 325518195674, 5711193510092, 101260078423336, 1811480526001238, 32657053453306929, 592701233703282882, 10820725155122336406, 198584549759713158048, 3661487133197990007534

Offset: 0

Author

Seiichi Manyama, Aug 02 2025

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A064063, A386763.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)/(1+3*x)^2)/x)

PARI

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

Formula

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

a(n) = (1/(n+1)) * [x^n] ( (1+3*x)^2 / (1-2*x) )^(n+1).

D-finite with recurrence 2*(n+1)*a(n) +(-31*n+29)*a(n-1) +90*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Aug 03 2025