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A144904
Coefficient of x^n in expansion of x/((1-x-x^3)*(1-x)^(n-1)), also diagonal of A144903.
22
0, 1, 2, 6, 21, 76, 280, 1045, 3937, 14938, 56993, 218414, 840090, 3241153, 12537263, 48604755, 188799962, 734631798, 2862843281, 11171582151, 43647688211, 170720728344, 668414462009, 2619400928928, 10273572796046, 40325085206853, 158393604268277
OFFSET
0,3
LINKS
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
Sequence in context: A116821 A116772 A131792 * A376791 A151287 A294822
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 24 2008
STATUS
approved