OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
FORMULA
a(n) = [x^n] x/((1-x-x^3)*(1-x)^(n-1)).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n-1)/3)} binomial(2*n-2*j-2, n+j-1).
a(n) = A099567(2*n, n). (End)
a(n) = binomial(2*(n-1), n-1)*hypergeom([1, (1-n)/3, (2-n)/3, 1-n/3], [1-n, 3/2-n, n], -27/4) for n > 0. - Stefano Spezia, Apr 06 2024
a(n) ~ 4^n/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 08 2024
MAPLE
A:= proc(n, k) coeftayl (x/ (1-x-x^3)/ (1-x)^(k-1), x=0, n) end:
a:= n-> A(n, n):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, n,
((27*n^3-150*n^2+195*n-12)*a(n-1)
-(66*n^3-382*n^2+492*n+124)*a(n-2)
+(27*n^3-156*n^2+201*n+48)*a(n-3)
-2*(2*n-7)*(3*n^2-7*n-2)*a(n-4))/((n-1)*(3*n^2-13*n+8)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jun 06 2013
MATHEMATICA
Table[Sum[Binomial[2*n-2*j-2, n+j-1], {j, 0, Floor[(n-1)/3]}], {n, 0, 40}] (* G. C. Greubel, Jul 27 2022 *)
PROG
(Magma)
A144904:= func< n | n eq 0 select 0 else (&+[Binomial(2*n-2*j-2, n+j-1): j in [0..Floor((n-1)/3)]]) >;
[A144904(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022
(SageMath)
def A144904(n): return sum(binomial(2*n-2*j-2, n+j-1) for j in (0..((n-1)//3)))
[A144904(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 24 2008
STATUS
approved