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

0,2

Links

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

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

Example

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.

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

Crossrefs

Row k=6 of A287318.

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

Sequence in context: A138914 A326220 A308129 * A286038 A276482 A202788

Adjacent sequences: A356255 A356256 A356257 * A356259 A356260 A356261

Keyword

nonn,easy,walk

Author

Dave R.M. Langers, Oct 12 2022

Status

approved