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 A121678 a(n) = [x^n] (1 + x*(1+x)^n )^(n+1). 3

%I #3 Mar 30 2012 18:36:59

%S 1,2,9,52,425,4236,49294,655096,9731313,159114880,2832245911,

%T 54400757016,1119436524947,24532373640334,569732648555295,

%U 13962373137304496,359767723241891425,9715902692094061488

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

%C a(n) is divisible by (n+1): a(n)/(n+1) = A121679(n).

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

%e At n=5, a(5) = [x^5] (1 + x*(1+x)^5)^6 = 4236, since

%e (1+x*(1+x)^5)^6 = 1 + 6*x + 45*x^2 + 230*x^3 + 1050*x^4 + 4236*x^5 +...

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

%Y Cf. A121679; variants: A121673-A121676, A121680.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 15 2006

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