A321615 Triangle read by rows: T(n,k) is the number of k X k integer matrices with sum of elements n, with no zero rows or columns, up to row and column permutation.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 6, 3, 1, 0, 1, 9, 13, 3, 1, 0, 1, 17, 38, 20, 3, 1, 0, 1, 23, 97, 82, 23, 3, 1, 0, 1, 36, 217, 311, 126, 24, 3, 1, 0, 1, 46, 453, 968, 624, 151, 24, 3, 1, 0, 1, 65, 868, 2825, 2637, 933, 162, 24, 3, 1, 0, 1, 80, 1585, 7394, 10098, 4942, 1132, 165, 24, 3, 1

Offset: 0

Author

Andrew Howroyd, Nov 14 2018

Keywords

• nonn
• tabl

Comments

Also the number of non-isomorphic multiset partitions of weight n with k parts and k vertices, where the weight of a multiset partition is the sum of sizes of its parts. - Gus Wiseman, Nov 18 2018

Examples

			Triangle begins:
    1
    0  1
    0  1    1
    0  1    2    1
    0  1    6    3    1
    0  1    9   13    3    1
    0  1   17   38   20    3    1
    0  1   23   97   82   23    3    1
    0  1   36  217  311  126   24    3    1
    0  1   46  453  968  624  151   24    3    1
    0  1   65  868 2825 2637  933  162   24    3    1
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Columns k=0..3 are A000007, A000012, A054974, A054975.

Row sums are A319616.

Cf. A007716, A048291, A054976, A057149, A057150, A104601, A120732, A318795, A320808.

Programs

Mathematica

(* See A318795 for M[m, n, k]. *)
T[n_, k_] := M[k, k, n] - 2 M[k, k-1, n] + M[k-1, k-1, n];
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 24 2018, from PARI *)

PARI

\\ See A318795 for M.
T(n, k) = if(k==0, n==0, M(k, k, n) - 2*M(k, k-1, n) + M(k-1, k-1, n));

PARI

\\ See A340652 for G.
T(n)={[Vecrev(p) | p<-Vec(1 + sum(k=1, n, y^k*(polcoef(G(k, n, n, y), k, y) - polcoef(G(k-1, n, n, y), k, y))))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 16 2024

Extensions

Column k=0 inserted by Andrew Howroyd, Jan 17 2024