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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.
2
1, 12, 396, 19920, 1281420, 96807312, 8175770064, 748315668672, 72729762868620, 7402621930738320, 781429888276676496, 84955810313787521472, 9463540456205136873936, 1075903653146632508721600, 124461755084172965028753600, 14615050011682746903615601920
OFFSET
0,2
LINKS
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
KEYWORD
nonn,easy,walk
AUTHOR
Dave R.M. Langers, Oct 12 2022
STATUS
approved