A378467 Expansion of (1/x) * Series_Reversion( x * (1 - x - x^2/(1 - x)^2) ).

1, 1, 3, 12, 53, 249, 1223, 6207, 32296, 171355, 923583, 5042840, 27834231, 155052721, 870594423, 4921968177, 27995045409, 160080985928, 919731472614, 5306779508096, 30737417720495, 178654274650097, 1041678247875531, 6091298104643577, 35714017347725474

Offset: 0

Author

Seiichi Manyama, Nov 27 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A001002, A001003.

Cf. A378462.

Programs

PARI

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

PARI

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

Formula

G.f.: exp( Sum_{k>=1} A378462(k) * x^k/k ).

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

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