A144904 Coefficient of x^n in expansion of x/((1-x-x^3)*(1-x)^(n-1)), also diagonal of A144903.

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

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

Cf. A099567, A144903.

Sequence in context: A116821 A116772 A131792 * A151287 A294822 A294823

Adjacent sequences: A144901 A144902 A144903 * A144905 A144906 A144907

Keyword

nonn,changed

Author

Alois P. Heinz, Sep 24 2008

Status

approved