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A263159 Number A(n,k) of lattice paths starting at {n}^k and ending when k or any component equals 0, using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A263159 #16 Apr 25 2020 14:47:42
%S A263159 1,1,1,1,1,1,1,3,1,1,1,7,13,1,1,1,15,157,63,1,1,1,31,2101,5419,321,1,
%T A263159 1,1,63,32461,717795,220561,1683,1,1,1,127,580693,142090291,328504401,
%U A263159 9763807,8989,1,1,1,255,11917837,39991899123,944362553521,172924236255,454635973,48639,1,1
%N A263159 Number A(n,k) of lattice paths starting at {n}^k and ending when k or any component equals 0, using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A263159 Alois P. Heinz, <a href="/A263159/b263159.txt">Antidiagonals n = 0..20, flattened</a>
%e A263159 Square array A(n,k) begins:
%e A263159   1, 1,    1,       1,            1,                1, ...
%e A263159   1, 1,    3,       7,           15,               31, ...
%e A263159   1, 1,   13,     157,         2101,            32461, ...
%e A263159   1, 1,   63,    5419,       717795,        142090291, ...
%e A263159   1, 1,  321,  220561,    328504401,     944362553521, ...
%e A263159   1, 1, 1683, 9763807, 172924236255, 7622403922836151, ...
%p A263159 s:= proc(n) option remember; `if`(n=0, {[]},
%p A263159       map(x-> [[x[], 0], [x[], 1]][], s(n-1)))
%p A263159     end:
%p A263159 b:= proc(l) option remember; `if`(l=[] or l[1]=0, 1,
%p A263159        add((p-> `if`(p[1]<0, 0, `if`(p[1]=0, 1, b(p)))
%p A263159        )(sort(l-x)), x=s(nops(l)) minus {[0$nops(l)]}))
%p A263159     end:
%p A263159 A:= (n, k)-> b([n$k]):
%p A263159 seq(seq(A(n,d-n), n=0..d), d=0..10);
%t A263159 g[k_] := Table[Reverse[IntegerDigits[n, 2]][[;;k]], {n, 2^k+1, 2^(k+1)-1}];
%t A263159 b[l_] := b[l] = If[l[[1]] == 0, 1, Sum[b[Sort[l - h]], {h, g[k]}]];
%t A263159 a[n_, k_] := If[n == 0 || k == 0 || k == 1, 1, b[Table[n, {k}]]];
%t A263159 Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Apr 25 2020, after _Alois P. Heinz_ in A115866 *)
%Y A263159 Columns k=0+1, 2-10 give: A000012, A001850, A115866, A263162, A263163, A263164, A263165, A263166, A263167, A263168.
%Y A263159 Rows n=0-1 give: A000012, A255047.
%Y A263159 Main diagonal gives A263160.
%Y A263159 Cf. A210472, A225094, A227578, A227655, A229142, A229345, A262809.
%K A263159 nonn,tabl
%O A263159 0,8
%A A263159 _Alois P. Heinz_, Oct 11 2015