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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.
3
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
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.
Sequence in context: A301945 A158843 A008793 * A015192 A012790 A273194
KEYWORD
nonn,walk
AUTHOR
Alois P. Heinz, Mar 08 2023
STATUS
approved