A397924 Number of nonnegative integer matrices with a total of n rows and columns with each row sum and column sum nonzero and at most 3, up to permutations of rows and columns.

1, 0, 3, 4, 16, 32, 100, 242, 710, 1906, 5628, 16166, 49197, 149238, 471665, 1501246, 4936440, 16416352, 56114886, 194327112, 689266460, 2478250388, 9101930092, 33886851758, 128597508191, 494569810826, 1935407461619, 7672504574678, 30904945481430, 126049107117078, 521740433396700, 2185664622577434, 9282632762832077

Offset

0,3

Comments

Equivalently, a(n) is the number of bicolored multigraphs on n unlabeled vertices with no isolated vertices and vertex degrees at most 3. The color classes are not interchangeable and edges may not connect vertices of the same color.

The inverse Euler transform gives A397917 with A397917(1) = 0.

Links

Table of n, a(n) for n=0..32.

Example

The a(3) = 4 matrices are:
   [1 1]  [2 1]  [1]  [2]
                 [1]  [1]
The a(4) = 16 matrices are:
  [1 1 1]  [1 0]  [1 1]  [1 1]  [2 0]  [2 0]  [2 0]  [2 1]
           [0 1]  [0 1]  [1 1]  [0 1]  [1 1]  [0 2]  [0 1]
    [1]
    [1]    [2 1]  [2 1]  [2 1]  [2 1]  [3 0]  [3 0]  [3 0]
    [1]    [1 0]  [1 1]  [0 2]  [1 2]  [0 1]  [0 2]  [0 3]

Crossrefs

Column k=3 of A397922.

Cf. A397917 (connected case), A397923, A397925 (if rows and columns can be zero).

Sequence in context: A123773 A290433 A251582 * A328773 A330693 A329541

Adjacent sequences: A397919 A397922 A397923 * A397925 A397926 A397927

Keyword

nonn,new

Author

Andrew Howroyd, Jul 15 2026

Status

approved