A006825 -id:A006825 - cp's OEIS frontend

A008326 Triangle read by rows: T(n,k) is the number of simple regular connected bipartite graphs with 2n nodes and degree k, (2 <= k <= n).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 4, 1, 1, 1, 13, 14, 4, 1, 1, 1, 38, 129, 41, 7, 1, 1, 1, 149, 1980, 1981, 157, 8, 1, 1, 1, 703, 62611, 304495, 62616, 725, 12, 1, 1, 1, 4132, 2806490, 78322915, 78322916, 2806508, 4196, 14, 1, 1, 1, 29579, 158937213, 27033154060, 147252447227, 27033154065, 158937367, 29817, 21, 1, 1

Offset: 2

Brendan McKay and Eric Rogoyski

This sequence can be derived from A133687 and A333159. In particular, if w(n) is the inverse Euler transform of column k of A133687 and s(n) is the inverse Euler transform of column k of A333159, then 2*T(2*n+1,k) = w(2*n+1) + s(2*n+1) and 2*T(2*n,k) = w(2*n) + s(2*n) - w(n) + T(n,k). - Andrew Howroyd, Apr 03 2020

			Triangle begins:
  1;
  1,   1;
  1,   1,    1;
  1,   2,    1,    1;
  1,   5,    4,    1,   1;
  1,  13,   14,    4,   1, 1;
  1,  38,  129,   41,   7, 1, 1;
  1, 149, 1980, 1981, 157, 8, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..154
B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.

Columns k=3..7 are A006823, A006824, A006825, A014385, A014387.
Row sums are in A008323.
Cf. A008327, A133687, A333159.

More terms from Eric Rogoyski, May 15 1997
Name clarified by Andrew Howroyd, Sep 05 2018
Terms a(55) and beyond from Andrew Howroyd, Apr 03 2020

A006824 Number of connected regular bipartite graphs of degree 4 with 2n nodes.

Original entry on oeis.org

1, 1, 4, 14, 129, 1980, 62611, 2806490, 158937213, 10773251972, 855658082615, 78558949838723, 8251166737356319, 982806379842257309, 131756174189661102281, 19748565896506014747623, 3289970433888731383271400, 605948436052375098046655323, 122796503871896458570959144266

Offset: 4

N. J. A. Sloane

CRC Handbook of Combinatorial Designs, 1996, p. 648.
I. A. Faradzev, Constructive enumeration of combinatorial objects, pp. 131-135 of Problèmes combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). Colloq. Internat. du C.N.R.S., No. 260, Centre Nat. Recherche Scient., Paris, 1978.
H. Gropp, On tactical configurations, regular bipartite graphs and (v,k,even)-designs, Discr. Math., 155 (1996), 81-98.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.
M. Meringer, Tables of Regular Graphs

Column 4 of A008326.

Cf. A006823, A006825, A333730.

Inverse Euler transform of A333730. - Andrew Howroyd, Apr 03 2020

Terms a(12) and beyond from Andrew Howroyd, Apr 03 2020

A333731 Number of simple 5-regular bipartite graphs with 2n nodes.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 4, 41, 1981, 304496, 78322916, 27033154065, 11934777413049, 6593485023089912, 4485517185017734885, 3707462300996644966161, 3680029088808704302571244, 4341813441626431900011624077, 6033239205199253768404631967656, 9792722283774379198364367866020815

Offset: 1

Andrew Howroyd, Apr 03 2020

nonn

Column k=5 of A008327.

Euler transform of A006825.

cp's OEIS Frontend

A008326 Triangle read by rows: T(n,k) is the number of simple regular connected bipartite graphs with 2n nodes and degree k, (2 <= k <= n).

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A006824 Number of connected regular bipartite graphs of degree 4 with 2n nodes.

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A333731 Number of simple 5-regular bipartite graphs with 2n nodes.

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