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

0,2

Links

Table of n, a(n) for n=0..18.

Index entries for reversions of series

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

Prog

(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);

Crossrefs

Cf. A064063, A386763.

Sequence in context: A295244 A379546 A389273 * A264072 A303056 A131655

Adjacent sequences: A386766 A386767 A386768 * A386770 A386771 A386772

Keyword

nonn

Author

Seiichi Manyama, Aug 02 2025

Status

approved