A360143 a(n) = Sum_{k=0..n} binomial(2*n+2*k,n-k).

1, 3, 13, 59, 271, 1250, 5775, 26696, 123423, 570576, 2637306, 12187755, 56312089, 260134905, 1201493926, 5548533913, 25619837773, 118283258215, 546041467522, 2520515546083, 11633752319476, 53693477980816, 247798435809211, 1143547904185879, 5277058908767419

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 - x * c(x)^4) ), where c(x) is the g.f. of A000108.

D-finite with recurrence +n*(n-7)*a(n) -(7*n-4)*(n-7)*a(n-1) +4*(n^2-13*n+17)*a(n-2) +(35*n^2-217*n+304)*a(n-3) -2*(n-2)*(7*n-29)*a(n-4) +4*(n-2)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Mar 12 2023

a(n) = binomial(2*n, n)*hypergeom([1, -n, 1/2+n, 1+n], [(1+n)/3, (2+n)/3, 1+n/3], -4/27). - Stefano Spezia, Jun 17 2025

a(n) ~ (31 + (3844 - 372*sqrt(93))^(1/3) + 2^(2/3)*(31*(31 + 3*sqrt(93)))^(1/3)) * ((5 + (187/2 - 9*sqrt(93)/2)^(1/3) + ((187 + 9*sqrt(93))/2)^(1/3))^n / (31*3^(n+1))). - Vaclav Kotesovec, Oct 19 2025

Maple

A360143 := proc(n)
    add(binomial(2*n+2*k, n-k), k=0..n) ;
end proc:
seq(A360143(n), n=0..70) ; # R. J. Mathar, Mar 12 2023

Mathematica

Table[Sum[Binomial[2n+2k, n-k], {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Jul 23 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(2*n+2*k, n-k));
(PARI) my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x*(2/(1+sqrt(1-4*x)))^4)))

Crossrefs

Cf. A001700, A032443, A108080, A360144.

Cf. A000108, A000984, A002057.

Sequence in context: A378463 A152594 A052984 * A262664 A151229 A333472

Adjacent sequences: A360140 A360141 A360142 * A360144 A360145 A360146

Keyword

nonn

Author

Seiichi Manyama, Jan 27 2023

Status

approved