OFFSET
0,8
COMMENTS
An interior point p = (p_1, ..., p_k) has k>0 components with 0<p_i<n for 1<=i<=k.
LINKS
Alois P. Heinz, Antidiagonals n = 0..20, flattened
EXAMPLE
A(n,0) = 1: [()].
A(0,k) = 1: [{0}^k].
A(1,1) = 1: [(1), (0)].
A(2,1) = 0, there is no path from (2) to (0) without interior points.
A(1,2) = 2: [(1,1), (0,1), (0,0)], [(1,1), (1,0), (0,0)].
A(1,3) = 6: [(1,1,1), (0,1,1), (0,0,1), (0,0,0)], [(1,1,1), (0,1,1), (0,1,0), (0,0,0)], [(1,1,1), (1,0,1), (0,0,1), (0,0,0)], [(1,1,1), (1,0,1), (1,0,0), (0,0,0)], [(1,1,1), (1,1,0), (0,1,0), (0,0,0)], [(1,1,1), (1,1,0), (1,0,0), (0,0,0)].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 2, 6, 24, 120, ...
1, 0, 2, 54, 1944, 99000, ...
1, 0, 2, 384, 132000, 79716000, ...
1, 0, 2, 2550, 8059800, 57010275000, ...
1, 0, 2, 16506, 471369024, 38606650125120, ...
MAPLE
b:= proc(n, l) option remember; local m; m:= nops(l);
`if`(m=0 or l[m]=0, 1, `if`(l[1]>0 and l[m]<n, 0,
add(`if`(l[i]=0, 0, b(n, sort(subsop(i=l[i]-1, l)))), i=1..m)))
end:
A:= (n, k)-> b(n, [n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
b[n_, l_] := b[n, l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[l[[1]] > 0 && l[[m]] < n, 0, Sum[If[l[[i]] == 0, 0, b[n, Sort[ReplacePart[l, i -> l[[i]] - 1]]]], {i, 1, m}]]] ]; a[n_, k_] := b[n, Array[n&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)
CROSSREFS
Main diagonal gives: A225111.
KEYWORD
AUTHOR
Alois P. Heinz, Apr 27 2013
STATUS
approved