A361297 Number of n-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin.

1, 2, 20, 996, 108136, 19784060, 5389230384, 2031493901304, 1009373201680848, 638377781979995244, 500510427096797296240, 476433596774288713285352, 541348750963243079098368768, 723928411313545718524263072248, 1125748074023593276830674831519936

Offset: 0

Author

Alois P. Heinz, Mar 08 2023

Keywords

• nonn
• walk

Comments

a(n) is a multiple of 2n for n>=1.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..212

Crossrefs

Main diagonal of A361397.

Cf. A005843, A303503.

Programs

Maple

b:= proc(n, l) option remember; add(add((h-> `if`(n<=
      add(v, v=h), 0, `if`(n=1, 1, `if`(h[-1]=0, 0,
        b(n-1, h)))))(sort(subsop(i=abs(l[i]+j), l))),
          j=[-1, 1]), i=1..nops(l))
    end:
a:= n-> `if`(n=0, 1, b(2*n, [0$n])):
seq(a(n), n=0..15);
# second Maple program:
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:
g:= proc(n, k) option remember; `if` (n<1, -1,
      -add(g(n-i, k)*(2*i)!*b(i, k)/i!^2, i=1..n))
    end:
a:= n-> abs(g(n$2)):
seq(a(n), n=0..15);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, Sum[b[n - j, i - 1]*Binomial[n, j]^2, {j, 0, n}]];
g[n_, k_] := g[n, k] = If [n < 1, -1, -Sum[g[n - i, k]*(2i)!* b[i, k]/i!^2, {i, 1, n}]];
a[n_] := Abs[g[n, n]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, May 27 2023, from 2nd Maple program *)

Formula

a(n) = A361397(n,n).

From Vaclav Kotesovec, Apr 23 2023: (Start)

a(n) ~ c * d^n * n^(2*n), where d = 1.138128465642... and c = 1.72802011936...

a(n) ~ A303503(n). (End)