A356258 Number of 6-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.

1, 12, 396, 19920, 1281420, 96807312, 8175770064, 748315668672, 72729762868620, 7402621930738320, 781429888276676496, 84955810313787521472, 9463540456205136873936, 1075903653146632508721600, 124461755084172965028753600, 14615050011682746903615601920

Offset: 0

Author

Dave R.M. Langers, Oct 12 2022

Keywords

• nonn
• easy
• walk

Examples

			a(1)=12, because twelve paths start at the origin, visit one of the adjacent vertices, and immediately return to the origin, resulting in 12 different paths of length 2n=2*1=2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..467

Crossrefs

Row k=6 of A287318.

1-5 dimensional analogs are A000984, A002894, A002896, A039699, A287317.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
a:= n-> (2*n)!*b(n, 6)/n!^2:
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 30 2023

Formula

E.g.f.: Sum_{n>=0} a(2*n) * x^(2*n)/(2*n)! = I_0(2*x)^6. (I = Modified Bessel function first kind).

a(n) = Sum_{h+i+j+k+l+m=n, 0<=h,i,j,k,l,m<=n} multinomial(2n [h,h,i,i,j,j,k,k,l,l,m,m]). - Shel Kaphan, Jan 29 2023