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A318191 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 we have abs(p_{i}-p_{(i mod k)+1}) <= 1 and the first component used is p_1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A318191 #27 Oct 22 2023 11:35:51
%S A318191 1,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,6,12,4,1,1,1,24,180,72,8,1,1,1,120,
%T A318191 4680,5400,432,16,1,1,1,720,187200,914400,162000,2592,32,1,1,1,5040,
%U A318191 10634400,296438400,178660800,4860000,15552,64,1,1,1,40320,813664800,162273628800,469551168000,34907788800,145800000,93312,128,1,1
%N A318191 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 we have abs(p_{i}-p_{(i mod k)+1}) <= 1 and the first component used is p_1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A318191 Alois P. Heinz, <a href="/A318191/b318191.txt">Antidiagonals n = 0..20, flattened</a>
%e A318191 A(2,2) = 2^2 = 4:
%e A318191                     (0,1)
%e A318191                    /     \
%e A318191   (2,2)-(1,2)-(1,1)       (0,0)
%e A318191                    \     /
%e A318191                     (1,0)
%e A318191 Square array A(n,k) begins:
%e A318191   1, 1,  1,     1,         1,             1,                   1, ...
%e A318191   1, 1,  1,     2,         6,            24,                 120, ...
%e A318191   1, 1,  2,    12,       180,          4680,              187200, ...
%e A318191   1, 1,  4,    72,      5400,        914400,           296438400, ...
%e A318191   1, 1,  8,   432,    162000,     178660800,        469551168000, ...
%e A318191   1, 1, 16,  2592,   4860000,   34907788800,     743761386086400, ...
%e A318191   1, 1, 32, 15552, 145800000, 6820487308800, 1178106009360998400, ...
%p A318191 b:= proc(l) option remember; (n-> `if`(n<2 or max(l[])=0, 1,
%p A318191       add(`if`(l[i]=0 or 1<abs(l[`if`(i=1, 0, i)-1]-l[i]+1)
%p A318191        or 1<abs(l[`if`(i=n, 0, i)+1]-l[i]+1), 0,
%p A318191       b(subsop(i=l[i]-1, l))), i=1..n)))(nops(l))
%p A318191     end:
%p A318191 A:= (n, k)-> `if`(k<2 or n=0, 1, b([n-1, n$k-1])):
%p A318191 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A318191 b[l_] := b[l] = With[{n = Length[l]}, If[n < 2 || Max[l ] == 0, 1, Sum[If[ l[[i]] == 0 ||1 < Abs[l[[If[i == 1, 0, i] - 1]] - l[[i]] + 1] || 1 < Abs[l[[If[i == n, 0, i] + 1]] - l[[i]] + 1], 0, b[ReplacePart[l, i -> l[[i]] - 1]]], {i, n}]]];
%t A318191 A[n_, k_] :=  If[k < 2 || n == 0, 1, b[Join[{n - 1}, Table[n, {k - 1}]]]];
%t A318191 Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 13 2020, after Maple *)
%Y A318191 Columns k=0+1, 2 give: A000012, A011782.
%Y A318191 Rows n=0-2 give: A000012, A000142(n-1) for n>0, A322782/n for n>0.
%Y A318191 Main diagonal gives A320443.
%Y A318191 Cf. A227655.
%K A318191 nonn,tabl
%O A318191 0,12
%A A318191 _Alois P. Heinz_, Jan 07 2019