A360083 a(n) = Sum_{k=0..n} binomial(5*k,n-k) * Catalan(k).

1, 1, 7, 35, 189, 1092, 6538, 40278, 253730, 1626858, 10582616, 69669273, 463319257, 3107941405, 21004392887, 142882885210, 977562617826, 6722361860888, 46438235933700, 322111000796428, 2242538435656450, 15665017062799230, 109761527468995102

Offset: 0

Author

Seiichi Manyama, Jan 25 2023

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A052709, A073155, A360076, A360082.

Cf. A000108, A360090.

Programs

Mathematica

Table[Sum[Binomial[5k,n-k]CatalanNumber[k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jul 13 2025 *)

PARI

a(n) = sum(k=0, n, binomial(5*k, n-k)*binomial(2*k, k)/(k+1));

PARI

my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*(1+x)^5)))

Formula

G.f. A(x) satisfies A(x) = 1/(1 - x * (1+x)^5 * A(x)).

G.f.: 2 / (1 + sqrt( 1 - 4*x*(1+x)^5 )).