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A335570 Number A(n,k) of n-step k-dimensional nonnegative 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 A335570 #27 Jul 22 2021 15:18:55
%S A335570 1,1,1,1,1,1,1,1,2,1,1,1,3,3,1,1,1,4,7,6,1,1,1,5,13,17,10,1,1,1,6,21,
%T A335570 40,47,20,1,1,1,7,31,81,136,125,35,1,1,1,8,43,146,325,496,333,70,1,1,
%U A335570 1,9,57,241,686,1433,1753,939,126,1,1,1,10,73,372,1315,3476,6473,6256,2597,252,1
%N A335570 Number A(n,k) of n-step k-dimensional nonnegative 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 A335570 Alois P. Heinz, <a href="/A335570/b335570.txt">Antidiagonals n = 0..60, flattened</a>
%F A335570 A(n,k) == 1 (mod k) for k >= 2.
%e A335570 A(2,2) = 3: [(0,0),(1,1),(2,2)], [(0,0),(1,1),(0,1)], [(0,0),(1,1),(1,0)].
%e A335570 Square array A(n,k) begins:
%e A335570   1,  1,   1,    1,    1,     1,     1,      1, ...
%e A335570   1,  1,   1,    1,    1,     1,     1,      1, ...
%e A335570   1,  2,   3,    4,    5,     6,     7,      8, ...
%e A335570   1,  3,   7,   13,   21,    31,    43,     57, ...
%e A335570   1,  6,  17,   40,   81,   146,   241,    372, ...
%e A335570   1, 10,  47,  136,  325,   686,  1315,   2332, ...
%e A335570   1, 20, 125,  496, 1433,  3476,  7525,  14960, ...
%e A335570   1, 35, 333, 1753, 6473, 18711, 46165, 102173, ...
%e A335570   ...
%p A335570 b:= proc(n, l) option remember; `if`(n=0, 1, b(n-1, map(x-> x+1, l))+add(
%p A335570      `if`(l[i]>0, b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..nops(l)))
%p A335570     end:
%p A335570 A:= (n, k)-> b(n, [0$k]):
%p A335570 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A335570 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 A335570 A[n_, k_] := b[n, Table[0, {k}]];
%t A335570 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 29 2021, after _Alois P. Heinz_ *)
%Y A335570 Columns k=0-10 give: A000012, A001405, A151265, A149424, A346225, A346226, A346227, A346228, A346229, A346230, A346231.
%Y A335570 Rows n=0+1,2-3 give: A000012, A000027(k+1), A002061(k+1).
%Y A335570 Main diagonal gives A335588.
%Y A335570 Cf. A340591.
%K A335570 nonn,tabl,walk
%O A335570 0,9
%A A335570 _Alois P. Heinz_, Jan 26 2021