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

%I A361297 #20 May 27 2023 06:54:44
%S A361297 1,2,20,996,108136,19784060,5389230384,2031493901304,1009373201680848,
%T A361297 638377781979995244,500510427096797296240,476433596774288713285352,
%U A361297 541348750963243079098368768,723928411313545718524263072248,1125748074023593276830674831519936
%N A361297 Number of n-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin.
%C A361297 a(n) is a multiple of 2n for n>=1.
%H A361297 Alois P. Heinz, <a href="/A361297/b361297.txt">Table of n, a(n) for n = 0..212</a>
%F A361297 a(n) = A361397(n,n).
%F A361297 From _Vaclav Kotesovec_, Apr 23 2023: (Start)
%F A361297 a(n) ~ c * d^n * n^(2*n), where d = 1.138128465642... and c = 1.72802011936...
%F A361297 a(n) ~ A303503(n). (End)
%p A361297 b:= proc(n, l) option remember; add(add((h-> `if`(n<=
%p A361297       add(v, v=h), 0, `if`(n=1, 1, `if`(h[-1]=0, 0,
%p A361297         b(n-1, h)))))(sort(subsop(i=abs(l[i]+j), l))),
%p A361297           j=[-1, 1]), i=1..nops(l))
%p A361297     end:
%p A361297 a:= n-> `if`(n=0, 1, b(2*n, [0$n])):
%p A361297 seq(a(n), n=0..15);
%p A361297 # second Maple program:
%p A361297 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A361297       add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
%p A361297     end:
%p A361297 g:= proc(n, k) option remember; `if` (n<1, -1,
%p A361297       -add(g(n-i, k)*(2*i)!*b(i, k)/i!^2, i=1..n))
%p A361297     end:
%p A361297 a:= n-> abs(g(n$2)):
%p A361297 seq(a(n), n=0..15);
%t A361297 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}]];
%t A361297 g[n_, k_] := g[n, k] = If [n < 1, -1, -Sum[g[n - i, k]*(2i)!* b[i, k]/i!^2, {i, 1, n}]];
%t A361297 a[n_] := Abs[g[n, n]];
%t A361297 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, May 27 2023, from 2nd Maple program *)
%Y A361297 Main diagonal of A361397.
%Y A361297 Cf. A005843, A303503.
%K A361297 nonn,walk
%O A361297 0,2
%A A361297 _Alois P. Heinz_, Mar 08 2023