A210472 Number A(n,k) of paths starting at {n}^k to a border position where one component equals 0 using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 33, 20, 1, 0, 1, 5, 196, 543, 70, 1, 0, 1, 6, 1305, 22096, 10497, 252, 1, 0, 1, 7, 9786, 1304045, 3323092, 220503, 924, 1, 0, 1, 8, 82201, 106478916, 1971644785, 574346824, 4870401, 3432, 1, 0
Offset: 0
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..24, flattened
Crossrefs
Programs
-
Maple
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); -
Mathematica
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 *)