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A229345 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by the same positive integer; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A229345 #29 Oct 07 2018 18:36:34
%S A229345 1,1,1,1,1,1,1,3,2,1,1,7,22,4,1,1,25,248,188,8,1,1,121,6506,11380,
%T A229345 1712,16,1,1,721,292442,2359348,577124,16098,32,1,1,5041,19450082,
%U A229345 1088626684,991365512,30970588,154352,64,1
%N A229345 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by the same positive integer; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A229345 Alois P. Heinz, <a href="/A229345/b229345.txt">Antidiagonals n = 0..20, flattened</a>
%e A229345 A(2,2) = 22: [(2,2),(1,1),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)], [(2,2),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(1,2),(1,0),(0,0)], [(2,2),(0,2),(0,1),(0,0)], [(2,2),(0,2),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(0,0)], [(2,2),(2,1),(1,1),(0,1),(0,0)], [(2,2),(2,1),(1,1),(1,0),(0,0)], [(2,2),(2,1),(0,1),(0,0)], [(2,2),(2,1),(2,0),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,0)], [(2,2),(2,0),(1,0),(0,0)], [(2,2),(2,0),(0,0)].
%e A229345 Square array A(n,k) begins:
%e A229345   1,  1,     1,        1,            1,                 1, ...
%e A229345   1,  1,     3,        7,           25,               121, ...
%e A229345   1,  2,    22,      248,         6506,            292442, ...
%e A229345   1,  4,   188,    11380,      2359348,        1088626684, ...
%e A229345   1,  8,  1712,   577124,    991365512,     4943064622568, ...
%e A229345   1, 16, 16098, 30970588, 453530591824, 25162900228200976, ...
%p A229345 b:= proc(l) option remember; local m; m:= nops(l);
%p A229345       `if`(m=0 or l[m]=0, 1,
%p A229345       `if`(m>1, add(b(l-[j$m]), j=1..l[1]), 0)+
%p A229345       add(add(b(sort(subsop(i=l[i]-j, l))), j=1..l[i]), i=1..m))
%p A229345     end:
%p A229345 A:= (n, k)-> b([n$k]):
%p A229345 seq(seq(A(n, d-n), n=0..d), d=0..10);  # _Alois P. Heinz_, Sep 24 2013
%t A229345 b[l_] := b[l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[m > 1, Sum[b[l - Array[j&, m]], {j, 1, l[[1]]}],  0] + Sum[Sum[b[Sort[ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, m}]]]; a[n_, k_] := b[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 A229345 Columns k=0-3 give: A000012, A011782, A132595(n+1), A229482.
%Y A229345 Rows n=0-2 give: A000012, A038507 (for k>1), A229465.
%Y A229345 Main diagonal gives: A229346.
%Y A229345 Cf. A060854, A227578, A227655, A225094, A210472, A229142, A262809, A263159.
%K A229345 nonn,tabl
%O A229345 0,8
%A A229345 _Alois P. Heinz_, Sep 24 2013