%I #5 Jun 12 2015 05:48:27

%S 1,1,2,8,50,402,3932,45075,588450,8580542,137799497,2410575026,

%T 45531000715,921946835474,19895218322982,455271977561120,

%U 11000793881924130,279648297003419318,7454931579222301709

%N a(n) = A121676(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)^3 )^5/5 = 250/5 = 50, since

%e (1 + x*(1+x)^3 )^5 = 1 + 5*x + 25*x^2 + 85*x^3 + 250*x^4 +...

%t Flatten[{1,Table[Sum[Binomial[n+1,k] * Binomial[(n-1)*k,n-k] / (n+1), {k,0,n+1}], {n, 1, 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. A121676; variants: A121673-A121675, A121678-A121680.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 15 2006