A360871 -id:A360871 - cp's OEIS frontend

A360870 Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes, no cut-points and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).

1, 1, 2, 1, 4, 1, 7, 2, 1, 10, 8, 2, 1, 14, 19, 11, 1, 18, 40, 48, 7, 1, 23, 77, 154, 70, 5, 1, 28, 132, 421, 392, 71, 1, 34, 217, 1008, 1638, 690, 35, 1, 40, 340, 2210, 5623, 4548, 767, 16, 1, 47, 510, 4477, 16745, 22657, 8594, 566, 1, 54, 742, 8557, 44698, 92844, 64716, 11247, 226

Offset: 2

Andrew Howroyd, Feb 25 2023

Columns k >= 3 correspond to the 2-connected graphs.

Terms may be computed using the tools geng, vcolg and multig in nauty with some additional processing to check the degrees of nodes.

			Triangle begins:
  1;
  1,  2;
  1,  4;
  1,  7,   2;
  1, 10,   8,    2;
  1, 14,  19,   11;
  1, 18,  40,   48,     7;
  1, 23,  77,  154,    70,     5;
  1, 28, 132,  421,   392,    71;
  1, 34, 217, 1008,  1638,   690,    35;
  1, 40, 340, 2210,  5623,  4548,   767,    16;
  1, 47, 510, 4477, 16745, 22657,  8594,   566;
  1, 54, 742, 8557, 44698, 92844, 64716, 11247, 226;
  ...

Brendan McKay and Adolfo Piperno, nauty and Traces.

Column 2 is A014616.
Row sums are A360882.
Row sums except first column are A360871.
Cf. A046752, A322115, A360862, A360866.

A360882 Number of unlabeled connected multigraphs with n edges, no cut-points and degree >= 3 at each node, loops allowed.

Original entry on oeis.org

0, 1, 3, 5, 10, 21, 45, 114, 330, 1045, 3623, 13545, 53597, 223085, 969159

Offset: 1

Andrew Howroyd, Feb 27 2023

			The a(3) = 3 multigraphs are:
  - a single vertex with 3 loops;
  - a triple edge;
  - a single edge with a loop at each vertex.

Cf. A002935, A360863, A360867, A360871.

Row sums of A360870.

a(n) = A360871(n) + 1 for n > 1.

cp's OEIS Frontend

A360870 Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes, no cut-points and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).

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