A381911 Expansion of (1/x) * Series_Reversion( x * (1-x) / B(x) ), where B(x) is the g.f. of A001764.

1, 2, 9, 55, 394, 3102, 25969, 226891, 2045342, 18883205, 177640462, 1696658418, 16408796013, 160366113609, 1581329919636, 15713344659359, 157187582466527, 1581676730708500, 15998326150898211, 162571286470135097, 1658893916098102321, 16991130941208846890

Offset: 0

Author

Seiichi Manyama, Mar 10 2025

Keywords

• nonn

Crossrefs

Cf. A381912, A381913.

Cf. A381817, A381914.

Cf. A001764.

Programs

Mathematica

Table[Sum[Binomial[n + 3*k + 1, k] * Binomial[2*n - k, n - k]/(n + 3*k + 1), {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 22 2025 *)

PARI

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

Formula

G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x*A(x)).

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