A360870 -id:A360870 - cp's OEIS frontend

A046752 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) loopless multigraphs with n edges on k nodes and degree >= 3 at each node, n >= 3, 2 <= k <= floor(2*n/3).

1, 1, 1, 1, 1, 2, 2, 1, 3, 5, 1, 4, 13, 4, 1, 6, 26, 24, 5, 1, 7, 47, 84, 38, 1, 9, 78, 233, 216, 23, 1, 11, 126, 557, 914, 314, 16, 1, 13, 188, 1193, 3077, 2270, 325, 1, 15, 276, 2355, 8915, 11592, 4015, 162, 1, 18, 391, 4370, 23008, 47079, 31443, 4495, 66

Offset: 3

N. J. A. Sloane

Original name: Triangle of number of homeomorphically irreducible stars with n edges and m nodes.

			Triangle begins:
  1;
  1;
  1,  1;
  1,  2,   2;
  1,  3,   5;
  1,  4,  13,    4;
  1,  6,  26,   24,    5;
  1,  7,  47,   84,   38;
  1,  9,  78,  233,  216,    23;
  1, 11, 126,  557,  914,   314,   16;
  1, 13, 188, 1193, 3077,  2270,  325;
  1, 15, 276, 2355, 8915, 11592, 4015, 162;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..93 (rows 3..18)
B. R. Heap, The enumeration of homeomorphically irreducible star graphs, J. Math. Phys., 7 (1966), 1582-1587.
Sean A. Irvine, Java program (github).

Row sums are A002935.
Diagonal sums are A360879.
Cf. A339160, A360866, A360870.

More terms from Sean A. Irvine, May 16 2020

Name edited and offset corrected by Andrew Howroyd, Feb 25 2023

A360871 Number of unlabeled nonseparable (or 2-connected) multigraphs with n edges and degree >= 3 at each node, loops allowed.

Original entry on oeis.org

0, 0, 2, 4, 9, 20, 44, 113, 329, 1044, 3622, 13544, 53596, 223084, 969158

Offset: 1

Andrew Howroyd, Feb 25 2023

A single-edge is considered to be nonseparable here.

			The a(3) = 2 multigraphs are:
  - a triple edge;
  - a single edge with a loop at each vertex.

Row sums of A360870.

Cf. A002935, A360863, A360867.

A360880 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) multigraphs with n edges and k nodes, loops allowed, n >= 1, 2 <= k <= n + 1.

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 6, 2, 1, 0, 9, 6, 3, 1, 0, 12, 14, 13, 4, 1, 0, 16, 28, 39, 22, 5, 1, 0, 20, 52, 112, 98, 39, 6, 1, 0, 25, 93, 281, 383, 236, 63, 8, 1, 0, 30, 152, 655, 1304, 1220, 515, 102, 9, 1, 0, 36, 242, 1408, 3980, 5418, 3512, 1077, 153, 11, 1, 0

Offset: 1

Andrew Howroyd, Feb 25 2023

			Triangle T(n,k) begins (n edges >= 1, k vertices >= 2):
   1;
   2,   0;
   4,   1,   0;
   6,   2,   1,    0;
   9,   6,   3,    1,    0;
  12,  14,  13,    4,    1,   0;
  16,  28,  39,   22,    5,   1,   0;
  20,  52, 112,   98,   39,   6,   1, 0;
  25,  93, 281,  383,  236,  63,   8, 1, 0;
  30, 152, 655, 1304, 1220, 515, 102, 9, 1, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..190 (rows 1..19)

Columns k=2..3 are A002620(n+1), A050531(n-3).
Row sums are A360881.
Cf. A290428, A322115, A339070, A339160, A360870.

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

A046752 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) loopless multigraphs with n edges on k nodes and degree >= 3 at each node, n >= 3, 2 <= k <= floor(2*n/3).

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A360871 Number of unlabeled nonseparable (or 2-connected) multigraphs with n edges and degree >= 3 at each node, loops allowed.

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A360880 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) multigraphs with n edges and k nodes, loops allowed, n >= 1, 2 <= k <= n + 1.

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

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