A121681 - OEIS

%I #5 Jun 12 2015 05:49:49

%S 1,1,4,19,131,1136,11670,138727,1864711,27843874,456081803,8114074563,

%T 155519173031,3189879446235,69629136671356,1609836360587087,

%U 39262941548917619,1006616998791629666,27044968746461571213

%N a(n) = A121680(n)/(n+1) = [x^n] (1 + x*(1+x)^(n+1) )^(n+1) / (n+1).

%F a(n) = Sum_{k=0..n+1} C(n+1,k) * C((n+1)*k,n-k) / (n+1).

%e At n=4, a(4) = [x^4] (1 + x*(1+x)^5 )^5 /5 = 655/5 = 131, since

%e (1 + x*(1+x)^5 )^5 = 1 + 5*x + 35*x^2 + 160*x^3 + 655*x^4 +...

%t Table[Sum[Binomial[n+1,k] * Binomial[(n+1)*k,n-k] / (n+1), {k,0,n+1}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 12 2015 *)

%o (PARI) a(n)=sum(k=0,n+1,binomial(n+1,k)*binomial((n+1)*k,n-k))/(n+1)

%Y Cf. A121680; variants: A121673-A121679.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 15 2006