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a(n) = [x^n] (1 + x*(1+x)^(n+1) )^n.
3

%I #5 Jun 12 2015 05:47:09

%S 1,1,7,43,371,3926,47622,654151,9999523,167557174,3046387103,

%T 59616689595,1247357472869,27747682830531,653192297754076,

%U 16206706672425167,422358302959175123,11526119161103900834

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

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

%e At n=4, a(4) = [x^4] (1 + x*(1+x)^5 )^4 = 371, since

%e (1 + x*(1+x)^5 )^4 = 1 + 4*x + 26*x^2 + 104*x^3 + 371*x^4 +...

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

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

%Y Cf. variants: A121673, A121674, A121676-A121680.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 15 2006