A024719 a(n) = (1/3)*(2 + Sum_{k=0..n} C(3k,k)).

1, 2, 7, 35, 200, 1201, 7389, 46149, 291306, 1853581, 11868586, 76380826, 493606726, 3201081874, 20821158234, 135776966762, 887393271311, 5811082966886, 38119865826421, 250447855600321, 1647729357535486, 10854207824989831, 71581930485576631, 472560922429972951, 3122648143126315651

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = Sum_{k=0..n} C(k-n, 2*n-2*k). - Paul Barry, Mar 15 2010

G.f.: (1-2*g)/((3*g-1)*(g^3-2*g^2+g-1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011

Conjecture: 2*n*(2*n-1)*a(n) + (-31*n^2 + 29*n - 6)*a(n-1) +3*(3*n-1)*(3*n-2)*a(n-2) = 0. - R. J. Mathar, Sep 29 2012

a(n) ~ 3^(3*n + 5/2)/(23*2^(2*n+1)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 07 2012

From G. C. Greubel, Jan 22 2026: (Start)

a(n) = a(n-1) + binomial(3*n-1, n-1), with a(0) = 1.

G.f.: (1/(3*(1-x)))*(2 + hypergeometric([1/3, 2/3]; [1/2]; 27*x/4)). (End)

Mathematica

Table[Sum[Binomial[k-n, 2n-2k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 07 2012 *)

Prog

(PARI) a(n)=sum(k=0, n, binomial(k-n, 2*(n-k)) ); \\ Joerg Arndt, May 04 2013
(Magma)
A024719:= func< n | n eq 0 select 1 else $$(n-1) +Binomial(3*n-1, n-1) >;
[A024719(n): n in [0..30]]; // G. C. Greubel, Jan 22 2026
(SageMath)
def A024719(n): return 1 if (n==0) else A024719(n-1) +binomial(3*n-1, n-1)
print([A024719(n) for n in range(31)]) # G. C. Greubel, Jan 22 2026

Crossrefs

Cf. A024718.

Sequence in context: A058941 A287327 A020066 * A086637 A172511 A214461

Adjacent sequences: A024716 A024717 A024718 * A024720 A024721 A024722

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from James Sellers, May 01 2000

Status

approved