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

0,2

Comments

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

Links

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

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)

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 *)

Crossrefs

Main diagonal of A361397.

Cf. A005843, A303503.

Sequence in context: A301945 A158843 A008793 * A015192 A012790 A273194

Adjacent sequences: A361294 A361295 A361296 * A361298 A361299 A361300

Keyword

nonn,walk

Author

Alois P. Heinz, Mar 08 2023

Status

approved