A008324 -id:A008324 - cp's OEIS frontend

A008327 Triangle read by rows: T(n,k) is the number of simple regular bipartite graphs with 2n nodes and degree k, (0 <= k <= n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 6, 4, 1, 1, 1, 1, 4, 14, 14, 4, 1, 1, 1, 1, 7, 41, 130, 41, 7, 1, 1, 1, 1, 8, 157, 1981, 1981, 157, 8, 1, 1, 1, 1, 12, 725, 62616, 304496, 62616, 725, 12, 1, 1, 1, 1, 14, 4196, 2806508, 78322916

Offset: 0

Brendan McKay

This sequence can be derived from A008326 by Euler transform. - Andrew Howroyd, Apr 03 2020

			Triangle begins:
  1,
  1, 1,
  1, 1, 1,
  1, 1, 1,   1,
  1, 1, 2,   1,    1,
  1, 1, 2,   2,    1,    1,
  1, 1, 4,   6,    4,    1,   1;
  1, 1, 4,  14,   14,    4,   1, 1;
  1, 1, 7,  41,  130,   41,   7, 1, 1;
  1, 1, 8, 157, 1981, 1981, 157, 8, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..189
B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.

Column k=0..5 are A000012, A000012, A002865, A008325, A333730, A333731.
Row sums are A008324.
Cf. A051031, A008326, A087114, A133687, A333159.

Column k is the Euler transform of column k of A008326. - Andrew Howroyd, Apr 03 2020

More terms from Eric Rogoyski, May 15 1997

Name clarified by Andrew Howroyd, Sep 05 2018

A008323 Number of simple connected regular bipartite graphs with 2n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 12, 34, 218, 4278, 431165, 162267174, 201636689352, 777816803938932, 9865957936943859185, 395886667549681686369527, 53716176608076643470380213991, 23524515269630339982914608683548933, 35682168849414944013547274439525153167248

Offset: 0

Brendan McKay

Brendan McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.

For n > 1, these are the row sums of triangle A008326.

Cf. A005142, A008324.

a(10) corrected and a(11) added by Brendan McKay, Sep 06 2018

a(0)=1 prepended and terms a(12) and beyond from Andrew Howroyd, Apr 03 2020

A087114 Number of regular bipartite simple graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 6, 1, 8, 1, 18, 1, 40, 1, 230, 1, 4296, 1, 431206, 1, 162267272, 1, 201636689771, 1, 777816803942186, 1, 9865957936943931964, 1, 395886667549681689591841, 1, 53716176608076643470621234239, 1, 23524515269630339982914646821899537, 1, 35682168849414944013547274452501768506834, 1

Offset: 0

Eric W. Weisstein, Aug 13 2003

nonn

A graph must be regular and bipartite to be a semisymmetric graph.

Eric Weisstein's World of Mathematics, Semisymmetric Graph

Cf. A008323, A008324, A008327.

a(2*n + 1) = 1, a(2*n) = A008324(n). - Andrew Howroyd, Sep 05 2018

a(10)-a(19) from Andrew Howroyd, Sep 05 2018

a(0)=1 prepended and a(20) onwards added by Andrew Howroyd, Feb 21 2024

A333732 Number of non-isomorphic n X n binary matrices with equal row and column sums up to permutation of rows and columns and transposition.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 18, 40, 230, 4296, 431206, 162267272, 201636689772, 777816803942188, 9865957936943931980, 395886667549681689592056, 53716176608076643470621240097, 23524515269630339982914646822137232, 35682168849414944013547274452501783251521

Offset: 0

Andrew Howroyd, Apr 03 2020

nonn

Number of simple regular bicolored graphs on 2n unlabeled nodes and allowing the color classes to be interchanged.

First differs from A008324 at n=12. See the note in A004066 by Sean A. Irvine for an explanation of why these two sequences are different.

Cf. A004066, A008324, A333160, A333681.

a(n) = (A333160(n) + A333681(n)) / 2.

a(n) >= A008324(n).

cp's OEIS Frontend

A008327 Triangle read by rows: T(n,k) is the number of simple regular bipartite graphs with 2n nodes and degree k, (0 <= k <= n).

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A008323 Number of simple connected regular bipartite graphs with 2n nodes.

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A087114 Number of regular bipartite simple graphs on n nodes.

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A333732 Number of non-isomorphic n X n binary matrices with equal row and column sums up to permutation of rows and columns and transposition.

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