A372004 G.f. A(x) satisfies A(x) = ( 1 + 9*x*A(x)*(1 + x*A(x)) )^(1/3).

1, 3, 3, 0, 9, 0, -63, 189, 0, -1944, 6399, 0, -72009, 245430, 0, -2921832, 10184130, 0, -125775585, 445134690, 0, -5641620192, 20188568790, 0, -260832419406, 941254831539, 0, -12342425759136, 44833549152825, 0, -594857401230510, 2172276845159733, 0

Offset

0,2

Links

Table of n, a(n) for n=0..32.

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 9^k * binomial(n/3+1/3,k) * binomial(k,n-k).

a(3*n+2) = 0 for n > 0.

a(n) = 9^n*binomial((n+1)/3, n)*hypergeom([(1-n)/2, -n/2], [2*(2-n)/3], 4/9)/(n+1). - Stefano Spezia, Apr 18 2024

D-finite with recurrence n*(n-1)*(2*n-11)*a(n) -108*(n-5)*(n-3)^2*a(n-3) -135*(n-5)*(n-8)*(2*n-5)*a(n-6)=0. - R. J. Mathar, Apr 22 2024

Maple

A372004 := proc(n)
    add(9^k*binomial((n+1)/3, k)*binomial(k, n-k), k=0..n)/(n+1) ;
end proc:
seq(A372004(n), n=0..60) ; # R. J. Mathar, Apr 22 2024

Prog

(PARI) a(n) = sum(k=0, n, 9^k*binomial(n/3+1/3, k)*binomial(k, n-k))/(n+1);

Crossrefs

Cf. A372005, A372006.

Sequence in context: A200701 A275408 A169670 * A341748 A120981 A298850

Adjacent sequences: A372001 A372002 A372003 * A372005 A372006 A372007

Keyword

sign

Author

Seiichi Manyama, Apr 15 2024

Status

approved