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.

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, 95, 1686, 11275, 4820, 1, 1, 1, 7, 161, 5796, 138740, 689146, 96030, 1, 1, 1, 8, 252, 16072, 1010385, 25876312, 76718466, 2766572, 1

Offset

0,9

Comments

A(n,k) counts generalized Tesler matrices. For the definition of Tesler matrices see A008608.

Links

Alois P. Heinz, Antidiagonals n = 0..15, flattened

Example

A(2,2) = 3: [1,1; 0,3], [2,0; 0,2], [0,2; 0,4].
Square array A(n,k) begins:
  1,   1,     1,      1,       1,       1, ...
  1,   1,     1,      1,       1,       1, ...
  1,   2,     3,      4,       5,       6, ...
  1,   7,    22,     50,      95,     161, ...
  1,  40,   351,   1686,    5796,   16072, ...
  1, 357, 11275, 138740, 1010385, 5244723, ...

Maple

b:= proc(n, i, l, k) option remember; (m-> `if`(m=0, 1,
      `if`(i=0, b(l[1]+k, m-1, subsop(1=NULL, l), k), add(
      b(n-j, i-1, subsop(i=l[i]+j, l), k), j=0..n))))(nops(l))
    end:
A:= (n, k)-> b(k, n-1, [0$(n-1)], k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

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 *)

Crossrefs

Columns k=0-2 give: A000012, A008608, A259919.

Rows n=0+1,2-3 give: A000012, A000027(k+1), A002412(k+1).

Sequence in context: A257493 A296526 A362377 * A112707 A196017 A343555

Adjacent sequences: A259841 A259842 A259843 * A259845 A259846 A259847

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 06 2015

Status

approved