A361829 a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(n*k,n-k).

1, 2, 10, 62, 486, 4482, 47106, 553226, 7152438, 100644194, 1527758136, 24839853326, 430045385424, 7888706328934, 152685931935634, 3106864307092950, 66253232332628166, 1476558925897693698, 34307420366092350048, 829217371825336147142

Offset: 0

Author

Seiichi Manyama, Mar 26 2023

Keywords

• nonn

Crossrefs

Main diagonal of A361830.

Cf. A099237, A361835.

Programs

Mathematica

Table[Sum[Binomial[2*k,k]*Binomial[n*k,n-k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 26 2023 *)

PARI

a(n) = sum(k=0, n, binomial(2*k, k)*binomial(n*k, n-k));

Formula

a(n) = [x^n] 1/sqrt(1 - 4*x*(1+x)^n).

log(a(n)) ~ n*(log(n) + (2*log(2) - 1)/log(n) - (1 - 1/log(n))*log(log(n) - 1)). - Vaclav Kotesovec, Mar 26 2023