A397924 - OEIS

%I #7 Jul 15 2026 19:40:48

%S 1,0,3,4,16,32,100,242,710,1906,5628,16166,49197,149238,471665,

%T 1501246,4936440,16416352,56114886,194327112,689266460,2478250388,

%U 9101930092,33886851758,128597508191,494569810826,1935407461619,7672504574678,30904945481430,126049107117078,521740433396700,2185664622577434,9282632762832077

%N 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.

%C 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.

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

%e The a(3) = 4 matrices are:

%e    [1 1]  [2 1]  [1]  [2]

%e                  [1]  [1]

%e The a(4) = 16 matrices are:

%e   [1 1 1]  [1 0]  [1 1]  [1 1]  [2 0]  [2 0]  [2 0]  [2 1]

%e            [0 1]  [0 1]  [1 1]  [0 1]  [1 1]  [0 2]  [0 1]

%e     [1]

%e     [1]    [2 1]  [2 1]  [2 1]  [2 1]  [3 0]  [3 0]  [3 0]

%e     [1]    [1 0]  [1 1]  [0 2]  [1 2]  [0 1]  [0 2]  [0 3]

%Y Column k=3 of A397922.

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

%K nonn,new

%O 0,3

%A _Andrew Howroyd_, Jul 15 2026