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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.

%I A225094 #34 Oct 07 2018 18:23:36
%S A225094 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,
%T A225094 720,99000,132000,2550,2,0,1,1,5040,6966000,79716000,8059800,16506,2,
%U A225094 0,1,1,40320,655678800,78928416000,57010275000,471369024,105840,2,0,1
%N 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.
%C A225094 An interior point p = (p_1, ..., p_k) has k>0 components with 0<p_i<n for 1<=i<=k.
%H A225094 Alois P. Heinz, <a href="/A225094/b225094.txt">Antidiagonals n = 0..20, flattened</a>
%e A225094 A(n,0) = 1: [()].
%e A225094 A(0,k) = 1: [{0}^k].
%e A225094 A(1,1) = 1: [(1), (0)].
%e A225094 A(2,1) = 0, there is no path from (2) to (0) without interior points.
%e A225094 A(1,2) = 2: [(1,1), (0,1), (0,0)], [(1,1), (1,0), (0,0)].
%e A225094 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)].
%e A225094 Square array A(n,k) begins:
%e A225094   1, 1, 1,     1,         1,              1, ...
%e A225094   1, 1, 2,     6,        24,            120, ...
%e A225094   1, 0, 2,    54,      1944,          99000, ...
%e A225094   1, 0, 2,   384,    132000,       79716000, ...
%e A225094   1, 0, 2,  2550,   8059800,    57010275000, ...
%e A225094   1, 0, 2, 16506, 471369024, 38606650125120, ...
%p A225094 b:= proc(n, l) option remember; local m; m:= nops(l);
%p A225094       `if`(m=0 or l[m]=0, 1, `if`(l[1]>0 and l[m]<n, 0,
%p A225094        add(`if`(l[i]=0, 0, b(n, sort(subsop(i=l[i]-1, l)))), i=1..m)))
%p A225094     end:
%p A225094 A:= (n, k)-> b(n, [n$k]):
%p A225094 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A225094 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 *)
%Y A225094 Columns k=0, 2-4 give: A000012, A040000, A060774, A225220.
%Y A225094 Rows n=0-4 give: A000012, A000142, A071798(k) (for k>0), A225096, A225221.
%Y A225094 Main diagonal gives: A225111.
%Y A225094 Cf. A089759 (unrestricted paths), A210472, A262809, A263159.
%K A225094 nonn,tabl,walk
%O A225094 0,8
%A A225094 _Alois P. Heinz_, Apr 27 2013