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
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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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
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Mathematica
Table[Sum[Binomial[2n+2k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jul 23 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(2*n+2*k, n-k));
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PARI
my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x*(2/(1+sqrt(1-4*x)))^4)))
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