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A227655 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_k) we have abs(p_{i}-p_{i+1}) <= 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A227655 #25 Jan 09 2019 14:50:08
%S A227655 1,1,1,1,1,1,1,2,1,1,1,6,4,1,1,1,24,44,8,1,1,1,120,896,320,16,1,1,1,
%T A227655 720,29392,33904,2328,32,1,1,1,5040,1413792,7453320,1281696,16936,64,
%U A227655 1,1,1,40320,93770800,2940381648,1897242448,48447504,123208,128,1,1
%N A227655 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_k) we have abs(p_{i}-p_{i+1}) <= 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A227655 Alois P. Heinz, <a href="/A227655/b227655.txt">Antidiagonals n = 0..18, flattened</a>
%e A227655 A(2,2) = 2^2 = 4:
%e A227655         (1,2)       (0,1)
%e A227655        /     \     /     \
%e A227655   (2,2)       (1,1)       (0,0)
%e A227655        \     /     \     /
%e A227655         (2,1)       (1,0)
%e A227655 Square array A(n,k) begins:
%e A227655   1, 1,  1,     1,        1,            1, ...
%e A227655   1, 1,  2,     6,       24,          120, ...
%e A227655   1, 1,  4,    44,      896,        29392, ...
%e A227655   1, 1,  8,   320,    33904,      7453320, ...
%e A227655   1, 1, 16,  2328,  1281696,   1897242448, ...
%e A227655   1, 1, 32, 16936, 48447504, 482913033152, ...
%p A227655 b:= proc(l) option remember; `if`({l[]}={0}, 1, add(
%p A227655       `if`(l[i]=0 or i>1 and 1<abs(l[i-1]-l[i]+1) or
%p A227655        i<nops(l) and 1<abs(l[i+1]-l[i]+1), 0,
%p A227655        b(subsop(i=l[i]-1, l))), i=1..nops(l)))
%p A227655     end:
%p A227655 A:= (n, k)-> `if`(k<2, 1, b([n$k])):
%p A227655 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A227655 b[l_] := b[l] = If[Union[l] == {0}, 1, Sum[If[l[[i]] == 0 || i>1 && 1 < Abs[l[[i-1]] - l[[i]] + 1] || i<Length[l] && 1<Abs[l[[i+1]] - l[[i]] + 1], 0, b[ReplacePart[l, i -> l[[i]]-1]]], {i, 1, Length[l]}]]; a[n_, k_] := If[k<2, 1, 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 A227655 Columns k=0+1, 2-10 give: A000012, A000079, A227665, A227666, A227667, A227668, A227669, A227670, A227671, A227672.
%Y A227655 Rows n=0-10 give: A000012, A000142, A227656, A227657, A227658, A227659, A227660, A227661, A227662, A227663, A227664.
%Y A227655 Main diagonal gives A227673.
%Y A227655 Cf. A262809, A263159, A318191.
%K A227655 nonn,tabl
%O A227655 0,8
%A A227655 _Alois P. Heinz_, Jul 19 2013