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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A326323 A257493 A296526 * A112707 A196017 A251660

Adjacent sequences:  A259841 A259842 A259843 * A259845 A259846 A259847

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 06 2015

STATUS

approved

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Last modified January 23 19:34 EST 2020. Contains 331175 sequences. (Running on oeis4.)