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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..15, flattened
Crossrefs
Programs
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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 *)
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