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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. 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

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

EXAMPLE

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, ...

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-Fran├žois Alcover, Dec 12 2013, translated from Maple *)

CROSSREFS

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: A287316 A322280 A331436 * A320080 A246106 A322836

Adjacent sequences:  A210469 A210470 A210471 * A210473 A210474 A210475

KEYWORD

nonn,tabl,walk

AUTHOR

Alois P. Heinz, Jan 22 2013

STATUS

approved

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Last modified January 26 23:05 EST 2020. Contains 331289 sequences. (Running on oeis4.)