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%I #8 Aug 10 2023 04:12:02
%S 1,1,5,33,250,2054,17800,160183,1482535,14022415,134943095,1317046306,
%T 13005842030,129708875695,1304588594925,13217663310305,
%U 134775670244250,1382019265706377,14242560597119165,147435736533094415,1532365596794307010
%N G.f. satisfies A(x) = 1 + x*A(x)^4 / (1 - x*A(x)^2).
%F a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(4*n-2*k,n-1-k) for n > 0.
%F D-finite with recurrence 3*n*(36653*n-48128)*(3*n-1)*(3*n+1)*a(n) +5*(-2160545*n^4 +5139476*n^3 -2463019*n^2 -1385144*n +913296)*a(n-1) +4*(-948403*n^4 +17991137*n^3 -77629283*n^2 +126107767*n -70578450)*a(n-2) +10*(n-3)*(599072*n^3 -5090881*n^2 +13501042*n -11263100)*a(n-3) -50*(6861*n-12886)*(n-3) *(n-4)*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Aug 10 2023
%p A364792 := proc(n)
%p if n = 0 then
%p 1;
%p else
%p add( binomial(n,k) * binomial(4*n-2*k,n-1-k),k=0..n-1) ;
%p %/n ;
%p end if ;
%p end proc:
%p seq(A364792(n),n=0..80); # _R. J. Mathar_, Aug 10 2023
%o (PARI) a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(4*n-2*k, n-1-k))/n);
%Y Cf. A002293, A243659, A349331, A364747, A364765.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Aug 08 2023