A165647 -id:A165647 - cp's OEIS frontend

A005177 Number of connected regular graphs with n nodes.

1, 1, 1, 1, 2, 2, 5, 4, 17, 22, 167, 539, 18979, 389436, 50314796, 2942198440, 1698517036411, 442786966115560, 649978211591600286, 429712868499646474880, 2886054228478618211088773, 8835589045148342277771518309, 152929279364927228928021274993215, 1207932509391069805495173301992815105, 99162609848561525198669168640159162918815

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

E. Friedman, Illustration of small graphs
Daniel R. Herber, Enhancements to the perfect matching approach for graph enumeration-based engineering challenges, Proceedings of the ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE 2020).
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
Markus Meringer, GENREG: A program for Connected Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Sep 23 2009]
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Regular Graph.

Regular simple graphs of any degree: this sequence (connected), A068932 (disconnected), A005176 (not necessarily connected), A275420 (multisets).
Connected regular graphs of any degree with girth at least g: this sequence (g=3), A186724 (g=4), A186725 (g=5), A186726 (g=6), A186727 (g=7), A186728 (g=8), A186729 (g=9).
Connected regular simple graphs: this sequence (any degree), A068934 (triangular array); specified degree k: A002851 (k=3), A006820 (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11). - Jason Kimberley, Nov 03 2011

a(n) = sum of the n-th row of A068934.
a(n) = A165647(n) - A165648(n).
This sequence is the inverse Euler transformation of A165647.

More terms from David Wasserman, Mar 08 2002
a(15) from Giovanni Resta, Feb 05 2009
Terms are sums of the output from M. Meringer's genreg software. To complete a(16) it was run by Jason Kimberley, Sep 23 2009
a(0)=1 (due to the empty graph being vacuously connected and regular) inserted by Jason Kimberley, Apr 11 2012
a(17)-a(21) from Andrew Howroyd, Mar 10 2020
a(22)-a(24) from Andrew Howroyd, May 19 2020

A275420 Triangle read by rows: T(n,k) = number of graphs with n nodes and k connected regular components.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 5, 5, 4, 2, 1, 1, 4, 9, 6, 4, 2, 1, 1, 17, 14, 12, 7, 4, 2, 1, 1, 22, 30, 19, 13, 7, 4, 2, 1, 1, 167, 56, 42, 22, 14, 7, 4, 2, 1, 1, 539, 224, 74, 47, 23, 14, 7, 4, 2, 1, 1, 18979, 785, 271, 87, 50, 24, 14, 7, 4, 2, 1, 1, 389436, 19783

Offset: 1

R. J. Mathar, Jul 27 2016

Multiset transformation of A005177.

The resulting graph has each component regular but may not be regular itself since different components can have different degrees. - Andrew Howroyd, May 20 2020

			      1
      1   1
      1   1   1
      2   2   1   1
      2   3   2   1   1
      5   5   4   2   1   1
      4   9   6   4   2   1   1
     17  14  12   7   4   2   1   1
     22  30  19  13   7   4   2   1   1
    167  56  42  22  14   7   4   2   1   1
    539 224  74  47  23  14   7   4   2   1   1
  18979 785 271  87  50  24  14   7   4   2   1   1

Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24)
Index entries for triangles generated by the Multiset Transformation

Cf. A005177 (1st column), A165647 (row sums).

T(n,1) = A005177(n).

T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1

G.f.: Product_{j>=1} (1-y*x^j)^(-A005177(j)). - Alois P. Heinz, Apr 13 2017

Name clarified by Andrew Howroyd, May 20 2020

A165648 Number of disconnected simple graphs on n vertices with each component regular.

Original entry on oeis.org

0, 1, 2, 4, 7, 13, 23, 41, 77, 149, 397, 1246, 21135, 430933, 51156773, 3044120326, 1704554902881, 446193132548644, 650868899188542416, 431014163502227412545, 2886915606822315071638459, 8841362446647790021087061250, 152946959203764346079534774815394, 1208238394473886999896406262410758886

Offset: 1

Jason Kimberley, Sep 23 2009

			The a(2)=1 graph is: 2K_1. The a(3)=2 graphs are: 3K_1, K_1+K_2. The a(4)=4 graphs are: 4K_1, 2K_1+K_2, K_1+K_3, 2K_2.

Cf. A005177, A165647.

A005177 = Cases[Import["https://oeis.org/A005177/b005177.txt", "Table"], {, }][[All, 2]]~Join~{0};
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[DivisorSum[j, # p[#]&] b[n - j], {j, 1, n}]/n]; b];
b = etr[A005177[[# + 1]]&];
a[n_] := b[n] - A005177[[n + 1]];
a /@ Range[17] (* Jean-François Alcover, Dec 02 2019 *)

a(n) = A165647(n) - A005177(n)

= Euler_transformation(A005177)(n) - A005177(n).

Terms a(18) and beyond from Andrew Howroyd, May 21 2020

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A005177 Number of connected regular graphs with n nodes.

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A275420 Triangle read by rows: T(n,k) = number of graphs with n nodes and k connected regular components.

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A165648 Number of disconnected simple graphs on n vertices with each component regular.

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