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A335588 Number of n-step n-dimensional nonnegative lattice walks starting at the origin and using steps that increment all components or decrement one component by 1.

%I A335588 #25 Jul 21 2021 10:52:27
%S A335588 1,1,3,13,81,686,7525,102173,1655241,31119382,665254791,15927737772,
%T A335588 422179410829,12275253219828,388591800808471,13309116622983421,
%U A335588 490515662121994785,19362705183912628838,815258217524407553989,36479395828632610279316,1729012534789121191076601
%N A335588 Number of n-step n-dimensional nonnegative lattice walks starting at the origin and using steps that increment all components or decrement one component by 1.
%H A335588 Vaclav Kotesovec, <a href="/A335588/b335588.txt">Table of n, a(n) for n = 0..65</a> (terms 0..55 from Alois P. Heinz)
%F A335588 a(n) = A335570(n,n).
%F A335588 a(n) == 1 (mod n) for n >= 2.
%e A335588 a(2) = 3: [(0,0),(1,1),(2,2)], [(0,0),(1,1),(0,1)], [(0,0),(1,1),(1,0)].
%p A335588 b:= proc(n, l) option remember; `if`(n=0, 1, b(n-1, map(x-> x+1, l))+add(
%p A335588      `if`(l[i]>0, b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..nops(l)))
%p A335588     end:
%p A335588 a:= n-> b(n, [0$n]):
%p A335588 seq(a(n), n=0..23);
%t A335588 b[n_, l_] := b[n, l] = If[n == 0, 1, b[n - 1, l + 1] + Sum[If[l[[i]] > 0, b[n - 1, Sort[ReplacePart[l, i -> l[[i]] - 1]]], 0], {i, 1, Length[l]}]];
%t A335588 a[n_] := b[n, Table[0, {n}]];
%t A335588 a /@ Range[0, 23] (* _Jean-François Alcover_, Jan 29 2021, after _Alois P. Heinz_ *)
%Y A335588 Main diagonal of A335570.
%K A335588 nonn,walk
%O A335588 0,3
%A A335588 _Alois P. Heinz_, Jan 26 2021