%I #42 Oct 24 2024 05:33:50

%S 1,4,22,128,777,4872,31330,205560,1370868,9266104,63343006,437183260,

%T 3042337215,21323543252,150395596016,1066637271424,7602188660799,

%U 54422262148632,391146728466980,2821396586367568,20417766975784066,148200184917042112

%N G.f. A(x) satisfies A(x) = (1 + x*A(x)^(3/4) / (1-x))^4.

%F a(n) = 4 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+4,k)/(3*k+4).

%F G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A307678.

%F a(n) ~ 9 * 31^(n + 1/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 3)). - _Vaclav Kotesovec_, Mar 29 2024

%F D-finite with recurrence 2*(n+2)*(2*n+3)*a(n) +(-55*n^2-74*n-15)*a(n-1) +6*(37*n^2-46*n-4)*a(n-2) -(295*n-319)*(n-3)*a(n-3) +124*(n-3)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Oct 24 2024

%p A370695 := proc(n)

%p 4*add(binomial(n-1,n-k)*binomial(3*k+4,k)/(3*k+4),k=0..n) ;

%p end proc:

%p seq(A370695(n),n=0..80) ; #_R. J. Mathar_, Oct 24 2024

%o (PARI) a(n) = 4*sum(k=0, n, binomial(n-1, n-k)*binomial(3*k+4, k)/(3*k+4));

%Y Cf. A045902, A062109, A371379, A371486, A371517.

%Y Cf. A270386, A307678, A371516.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 27 2024