A024720 a(n) = (1/4)*(3 + Sum_{k=0..n} C(4k,k)).

1, 2, 9, 64, 519, 4395, 38044, 334054, 2963629, 26499449, 238414581, 2155749364, 19572882981, 178326272881, 1629509263831, 14928031562011, 137059765831906, 1260847661188318, 11618870102584178, 107234108018545278, 991063143571588858, 9170871822844253578, 84959230298325555763, 787875497763596082613

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f.: (2*g-3)*g^4/((3*g-4)*(1-g+g^4)) where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011

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

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

a(n) = a(n-1) + binomial(4*n-1, n-1), with a(0) = 1.

G.f.: (1/(4*(1-x)))*(3 + hypergeometric([1/4, 1/2, 3/4], [1/3, 2/3], 256*x/27)). (End)

Mathematica

A024720[n_]:= A024720[n]= If[n==0, 1, A024720[n-1] +Binomial[4*n-1, n-1]];
Table[A024720[n], {n, 0, 30}] (* G. C. Greubel, Jan 22 2026 *)

Prog

(PARI) a(n) = (3+sum(k=0, n, binomial(4*k, k)))/4; \\ Michel Marcus, May 10 2020
(Magma)
A024720:= func< n | n eq 0 select 1 else $$(n-1) + Binomial(4*n-1, n-1) >;
[A024720(n): n in [0..30]]; // G. C. Greubel, Jan 22 2026
(SageMath)
def A024720(n): return 1 if (n==0) else A024720(n-1) +binomial(4*n-1, n-1)
print([A024720(n) for n in range(31)]) # G. C. Greubel, Jan 22 2026

Crossrefs

Cf. A002293, A024718.

Sequence in context: A074181 A052513 A216839 * A289717 A381908 A094100

Adjacent sequences: A024717 A024718 A024719 * A024721 A024722 A024723

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from James Sellers, May 01 2000

More terms added by G. C. Greubel, Jan 22 2026

Status

approved