A370695 G.f. A(x) satisfies A(x) = (1 + x*A(x)^(3/4) / (1-x))^4.

1, 4, 22, 128, 777, 4872, 31330, 205560, 1370868, 9266104, 63343006, 437183260, 3042337215, 21323543252, 150395596016, 1066637271424, 7602188660799, 54422262148632, 391146728466980, 2821396586367568, 20417766975784066, 148200184917042112

Offset: 0

Author

Seiichi Manyama, Mar 27 2024

Keywords

• nonn

Crossrefs

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

Cf. A270386, A307678, A371516.

Programs

Maple

A370695 := proc(n)
    4*add(binomial(n-1,n-k)*binomial(3*k+4,k)/(3*k+4),k=0..n) ;
end proc:
seq(A370695(n),n=0..80) ; #R. J. Mathar, Oct 24 2024

PARI

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

Formula

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

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

a(n) ~ 9 * 31^(n + 1/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 3)). - Vaclav Kotesovec, Mar 29 2024

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