A225094 Number A(n,k) of lattice paths without interior points from {n}^k to {0}^k using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 6, 2, 0, 1, 1, 24, 54, 2, 0, 1, 1, 120, 1944, 384, 2, 0, 1, 1, 720, 99000, 132000, 2550, 2, 0, 1, 1, 5040, 6966000, 79716000, 8059800, 16506, 2, 0, 1, 1, 40320, 655678800, 78928416000, 57010275000, 471369024, 105840, 2, 0, 1
Offset: 0
Comments
Examples
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, ...Links
Crossrefs
Programs
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] 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 *)