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A161799 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^2)^3.

%I A161799 #11 May 12 2022 17:01:56
%S A161799 1,3,12,61,345,2085,13182,86106,576543,3936029,27294390,191722887,
%T A161799 1361291244,9754412169,70447946556,512278417176,3747570671685,
%U A161799 27561220671408,203657352324178,1511270129552163,11257532921742528
%N A161799 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^2)^3.
%H A161799 G. C. Greubel, <a href="/A161799/b161799.txt">Table of n, a(n) for n = 0..1000</a>
%H A161799 Vaclav Kotesovec, <a href="/A161799/a161799.txt">Recurrence</a>
%F A161799 a(n) = Sum_{k=0..n} C(3*n-2*k+2,k)/(n-k+1) * C(n+k-1,n-k).
%F A161799 Let A(x)^m = Sum_{n>=0} a(n,m)*x^n then
%F A161799 a(n,m) = Sum_{k=0..n} C(3*n-2*k+3*m-1,k)*m/(n-k+m) * C(n+k-1,n-k).
%F A161799 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 = 1.56703431595354192843152170651865561188... - _Vaclav Kotesovec_, Sep 18 2013
%p A161799 A161799 := proc(n)
%p A161799     local s,t ;
%p A161799     s := 2 ;
%p A161799     t := 3;
%p A161799     add( binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k) /(n-k+1) ,k=0..n) ;
%p A161799 end proc:
%p A161799 seq(A161799(n),n=0..40) ; # _R. J. Mathar_, May 12 2022
%t A161799 Table[Sum[Binomial[3*n-2*k+2,k]/(n-k+1)*Binomial[n+k-1,n-k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Sep 18 2013 *)
%o A161799 (PARI) {a(n,m=1)=sum(k=0,n,binomial(3*n-2*k+3*m-1,k)*m/(n-k+m)*binomial(n+k-1,n-k))}
%Y A161799 Cf. A161797, A161798.
%K A161799 nonn
%O A161799 0,2
%A A161799 _Paul D. Hanna_, Jun 19 2009