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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
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
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
Columns k=0..3 are A000007, A000012, A054974, A054975.
Row sums are A319616.
Sequence in context: A291584 A352451 A349618 * A011126 A266854 A326882
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Nov 14 2018
EXTENSIONS
Column k=0 inserted by Andrew Howroyd, Jan 17 2024
STATUS
approved

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Last modified June 12 12:44 EDT 2024. Contains 373331 sequences. (Running on oeis4.)