A121680 - OEIS

%I #5 Oct 02 2020 09:49:10

%S 1,2,12,76,655,6816,81690,1109816,16782399,278438740,5016899833,

%T 97368894756,2021749249403,44658312247290,1044437050070340,

%U 25757381769393392,667470006331599523,18119105978249333988

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

%C a(n) is divisible by (n+1): a(n)/(n+1) = A121681(n).

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

%e At n=4, a(4) = [x^4] (1 + x*(1+x)^5 )^5 = 655, 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], {k,0,n+1}], {n,0,20}] (* _Vaclav Kotesovec_, Oct 02 2020 *)

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

%Y Cf. A121681; variants: A121673-A121679.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 15 2006