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

Author

Alois P. Heinz, Sep 24 2008

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A099567, A144903.

Programs

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

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 *)

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

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