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A210472
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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.
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12
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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
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OFFSET
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0,8
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LINKS
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EXAMPLE
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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, ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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