0,8

Alois P. Heinz, Antidiagonals n = 0..24, flattened

A(0,3) = 1: [(0,0,0)].

A(1,1) = 1: [(1), (0)].

A(1,2) = 2: [(1,1), (0,1)], [(1,1), (1,0)].

A(1,3) = 3: [(1,1,1), (0,1,1)], [(1,1,1), (1,0,1)], [(1,1,1), (1,1,0)].

A(2,1) = 1: [(2), (1), (0)].

A(2,2) = 6: [(2,2), (1,2), (0,2)], [(2,2), (1,2), (1,1), (0,1)], [(2,2), (1,2), (1,1), (1,0)], [(2,2), (2,1), (1,1), (0,1)], [(2,2), (2,1), (1,1), (1,0)], [(2,2), (2,1), (2,0)].

Square array A(n,k) begins:

0, 1, 1, 1, 1, 1, ...

0, 1, 2, 3, 4, 5, ...

0, 1, 6, 33, 196, 1305, ...

0, 1, 20, 543, 22096, 1304045, ...

0, 1, 70, 10497, 3323092, 1971644785, ...

0, 1, 252, 220503, 574346824, 3617739047205, ...

b:= proc() option remember; `if`(nargs=0, 0, `if`(args[1]=0, 1,

add(b(sort(subsop(i=args[i]-1, [args]))[]), i=1..nargs)))

end:

A:= (n, k)-> b(n$k):

seq(seq(A(n, d-n), n=0..d), d=0..10);

b[] = 0; b[args__] := b[args] = If[First[{args}] == 0, 1, Sum[b @@ Sort[ReplacePart[{args}, i -> {args}[[i]] - 1]], {i, 1, Length[{args}]}]]; a[n_, k_] := b @@ Array[n&, k]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

Columns k=0-4 give: A000004, A000012, A000984, A209245, A209288.

Rows n=0-3 give: A057427, A001477, A093964, A210486.

Main diagonal gives A276490.

Cf. A089759 (unrestricted paths), A225094, A262809, A263159.

Sequence in context: A331436 A343097 A343095 * A320080 A246106 A322836

Adjacent sequences: A210469 A210470 A210471 * A210473 A210474 A210475

nonn,tabl,walk

Alois P. Heinz, Jan 22 2013

approved