A356258 - OEIS

%I #43 Mar 10 2023 08:59:53

%S 1,12,396,19920,1281420,96807312,8175770064,748315668672,

%T 72729762868620,7402621930738320,781429888276676496,

%U 84955810313787521472,9463540456205136873936,1075903653146632508721600,124461755084172965028753600,14615050011682746903615601920

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

%H Alois P. Heinz, <a href="/A356258/b356258.txt">Table of n, a(n) for n = 0..467</a>

%F 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).

%F 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

%e 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.

%p b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,

%p       add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))

%p     end:

%p a:= n-> (2*n)!*b(n, 6)/n!^2:

%p seq(a(n), n=0..15);  # _Alois P. Heinz_, Jan 30 2023

%Y Row k=6 of A287318.

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

%K nonn,easy,walk

%O 0,2

%A _Dave R.M. Langers_, Oct 12 2022