A361829 - OEIS

%I #14 Mar 26 2023 10:25:20

%S 1,2,10,62,486,4482,47106,553226,7152438,100644194,1527758136,

%T 24839853326,430045385424,7888706328934,152685931935634,

%U 3106864307092950,66253232332628166,1476558925897693698,34307420366092350048,829217371825336147142

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

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

%F 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

%t Table[Sum[Binomial[2*k,k]*Binomial[n*k,n-k], {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 26 2023 *)

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

%Y Main diagonal of A361830.

%Y Cf. A099237, A361835.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 26 2023