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A259844 Number A(n,k) of n X n upper triangular matrices (m_{i,j}) of nonnegative integers with k = Sum_{j=h..n} m_{h,j} - Sum_{i=1..h-1} m_{i,h} for all h in {1,...,n}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A259844 #19 Sep 24 2018 12:47:16
%S A259844 1,1,1,1,1,1,1,1,2,1,1,1,3,7,1,1,1,4,22,40,1,1,1,5,50,351,357,1,1,1,6,
%T A259844 95,1686,11275,4820,1,1,1,7,161,5796,138740,689146,96030,1,1,1,8,252,
%U A259844 16072,1010385,25876312,76718466,2766572,1
%N A259844 Number A(n,k) of n X n upper triangular matrices (m_{i,j}) of nonnegative integers with k = Sum_{j=h..n} m_{h,j} - Sum_{i=1..h-1} m_{i,h} for all h in {1,...,n}; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A259844 A(n,k) counts generalized Tesler matrices.  For the definition of Tesler matrices see A008608.
%H A259844 Alois P. Heinz, <a href="/A259844/b259844.txt">Antidiagonals n = 0..15, flattened</a>
%e A259844 A(2,2) = 3: [1,1; 0,3], [2,0; 0,2], [0,2; 0,4].
%e A259844 Square array A(n,k) begins:
%e A259844   1,   1,     1,      1,       1,       1, ...
%e A259844   1,   1,     1,      1,       1,       1, ...
%e A259844   1,   2,     3,      4,       5,       6, ...
%e A259844   1,   7,    22,     50,      95,     161, ...
%e A259844   1,  40,   351,   1686,    5796,   16072, ...
%e A259844   1, 357, 11275, 138740, 1010385, 5244723, ...
%p A259844 b:= proc(n, i, l, k) option remember; (m-> `if`(m=0, 1,
%p A259844       `if`(i=0, b(l[1]+k, m-1, subsop(1=NULL, l), k), add(
%p A259844       b(n-j, i-1, subsop(i=l[i]+j, l), k), j=0..n))))(nops(l))
%p A259844     end:
%p A259844 A:= (n, k)-> b(k, n-1, [0$(n-1)], k):
%p A259844 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A259844 b[n_, i_, l_List, k_] := b[n, i, l, k] = Function[{m}, If[m == 0, 1, If[i == 0, b[l[[1]] + k, m-1, ReplacePart[l, 1 -> Sequence[]], k], Sum[b[n-j, i-1, ReplacePart[l, i -> l[[i]]+j], k], {j, 0, n}]]]][Length[l]]; A[n_, k_] := b[k, n-1, Array[0&, n-1], k]; A[0, _] = A[_, 0] = 1; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Jul 15 2015, after _Alois P. Heinz_ *)
%Y A259844 Columns k=0-2 give: A000012, A008608, A259919.
%Y A259844 Rows n=0+1,2-3 give: A000012, A000027(k+1), A002412(k+1).
%K A259844 nonn,tabl
%O A259844 0,9
%A A259844 _Alois P. Heinz_, Jul 06 2015