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A340591 Number A(n,k) of n*(k+1)-step k-dimensional nonnegative closed lattice walks starting at the origin and using steps that increment all components or decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A340591 #18 Jan 26 2021 21:01:00
%S A340591 1,1,1,1,1,1,1,2,2,1,1,6,16,5,1,1,24,288,192,14,1,1,120,9216,24444,
%T A340591 2816,42,1,1,720,460800,7303104,2738592,46592,132,1,1,5040,33177600,
%U A340591 4234233600,8204167296,361998432,835584,429,1,1,40320,3251404800,4223111040000,59027412643200,11332298092032,53414223552,15876096,1430,1
%N A340591 Number A(n,k) of n*(k+1)-step k-dimensional nonnegative closed lattice walks starting at the origin and using steps that increment all components or decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A340591 Alois P. Heinz, <a href="/A340591/b340591.txt">Antidiagonals n = 0..24, flattened</a>
%e A340591 Square array A(n,k) begins:
%e A340591   1,  1,     1,         1,              1,                   1, ...
%e A340591   1,  1,     2,         6,             24,                 120, ...
%e A340591   1,  2,    16,       288,           9216,              460800, ...
%e A340591   1,  5,   192,     24444,        7303104,          4234233600, ...
%e A340591   1, 14,  2816,   2738592,     8204167296,      59027412643200, ...
%e A340591   1, 42, 46592, 361998432, 11332298092032, 1052109889288796160, ...
%p A340591 b:= proc(n, l) option remember; `if`(n=0, 1, (k-> add(
%p A340591      `if`(l[i]>0, b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..k)+
%p A340591      `if`(add(i, i=l)+k<n, b(n-1, map(x-> x+1, l)), 0))(nops(l)))
%p A340591     end:
%p A340591 A:= (n, k)-> b(k*n+n, [0$k]):
%p A340591 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A340591 b[n_, l_] := b[n, l] = If[n == 0, 1, Function[k, Sum[
%t A340591   If[l[[i]]>0, b[n-1, Sort[ReplacePart[l, i -> l[[i]]-1]]], 0], {i, 1, k}]+
%t A340591   If[Sum[i, {i, l}] + k < n, b[n - 1, Map[#+1&, l]], 0]][Length[l]]];
%t A340591 A[n_, k_] := b[k*n + n, Table[0, {k}]];
%t A340591 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Jan 26 2021, after _Alois P. Heinz_ *)
%Y A340591 Columns k=0-3 give: A000012, A000108, A006335, A340540.
%Y A340591 Rows n=0-2 give: A000012, A000142, |A055546|.
%Y A340591 Main diagonal gives A340590.
%Y A340591 Cf. A335570.
%K A340591 nonn,tabl,walk
%O A340591 0,8
%A A340591 _Alois P. Heinz_, Jan 12 2021