A378425 Expansion of (1/x) * Series_Reversion( x / (1 + x + x^2 * (1 + x)^3) ).

1, 1, 2, 7, 24, 82, 297, 1121, 4317, 16900, 67185, 270480, 1100122, 4513809, 18661618, 77666327, 325117967, 1368001765, 5782686120, 24545144206, 104573104040, 447036252525, 1916918691196, 8243075111450, 35538551601880, 153584392913986, 665201585797986, 2887012910233897

Offset: 0

Author

Seiichi Manyama, Nov 25 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A036765, A378426.

Cf. A300048, A378427.

Cf. A378406.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^2*(1+x)^3))/x)

PARI

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

Formula

G.f.: exp( Sum_{k>=1} A378406(k) * x^k/k ).

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

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