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%I #8 Feb 12 2017 21:03:04
%S 1,2,9,46,262,1590,10081,65986,442518,3024772,20996141,147603198,
%T 1048747751,7519252606,54332565330,395264527626,2892666314150,
%U 21281120904168,157299607827727,1167582500757800,8699515577902203
%N G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^3)^2.
%H G. C. Greubel, <a href="/A161798/b161798.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="/A161798/a161798.txt">Recurrence</a>
%F a(n) = Sum_{k=0..n} C(2*n-k+1,k)/(n-k+1) * C(n+2*k-1,n-k).
%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n then
%F a(n,m) = Sum_{k=0..n} C(2*n-k+m,k)*m/(n-k+m) * C(n+2*k-1,n-k).
%F a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 8.01957328653868383... is the root of the equation 3125 + 22356*d - 162432*d^2 - 361584*d^3 - 326592*d^4 + 46656*d^5 = 0 and c = 1.216730444416766043545857948227854793382399566... - _Vaclav Kotesovec_, Sep 18 2013
%t Table[Sum[Binomial[2*n-k+1,k]/(n-k+1)*Binomial[n+2*k-1,n-k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Sep 18 2013 *)
%o (PARI) {a(n,m=1)=sum(k=0,n,binomial(2*n-k+m,k)*m/(n-k+m)*binomial(n+2*k-1,n-k))}
%Y Cf. A161797, A161799.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 19 2009
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