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%I #37 Oct 07 2018 18:31:14
%S 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,
%T 1305,22096,10497,252,1,0,1,7,9786,1304045,3323092,220503,924,1,0,1,8,
%U 82201,106478916,1971644785,574346824,4870401,3432,1,0
%N 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.
%H Alois P. Heinz, <a href="/A210472/b210472.txt">Antidiagonals n = 0..24, flattened</a>
%e A(0,3) = 1: [(0,0,0)].
%e A(1,1) = 1: [(1), (0)].
%e A(1,2) = 2: [(1,1), (0,1)], [(1,1), (1,0)].
%e A(1,3) = 3: [(1,1,1), (0,1,1)], [(1,1,1), (1,0,1)], [(1,1,1), (1,1,0)].
%e A(2,1) = 1: [(2), (1), (0)].
%e 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)].
%e Square array A(n,k) begins:
%e 0, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 0, 1, 6, 33, 196, 1305, ...
%e 0, 1, 20, 543, 22096, 1304045, ...
%e 0, 1, 70, 10497, 3323092, 1971644785, ...
%e 0, 1, 252, 220503, 574346824, 3617739047205, ...
%p b:= proc() option remember; `if`(nargs=0, 0, `if`(args[1]=0, 1,
%p add(b(sort(subsop(i=args[i]-1, [args]))[]), i=1..nargs)))
%p end:
%p A:= (n, k)-> b(n$k):
%p seq(seq(A(n, d-n), n=0..d), d=0..10);
%t 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 *)
%Y Columns k=0-4 give: A000004, A000012, A000984, A209245, A209288.
%Y Rows n=0-3 give: A057427, A001477, A093964, A210486.
%Y Main diagonal gives A276490.
%Y Cf. A089759 (unrestricted paths), A225094, A262809, A263159.
%K nonn,tabl,walk
%O 0,8
%A _Alois P. Heinz_, Jan 22 2013
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