A185315 -id:A185315 - cp's OEIS frontend

A005176 Number of regular graphs with n unlabeled nodes.

1, 1, 2, 2, 4, 3, 8, 6, 22, 26, 176, 546, 19002, 389454, 50314870, 2942198546, 1698517037030, 442786966117636, 649978211591622812, 429712868499646587714, 2886054228478618215888598, 8835589045148342277802657274, 152929279364927228928025482936226, 1207932509391069805495173417972533120, 99162609848561525198669168653641835566774

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
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Jennifer M. Larson, Cheating Because They Can: Social Networks and Norm Violators, 2014. See Footnote 11.
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.

Not necessarily connected simple regular graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Simple regular graphs of any degree: A005177 (connected), A068932 (disconnected), this sequence (not necessarily connected).
Not necessarily connected regular simple graphs with girth at least g: this sequence (g=3), A185314 (g=4), A185315 (g=5), A185316 (g=6), A185317 (g=7), A185318 (g=8), A185319 (g=9).
Cf. A295193.

a(n) = A005177(n) + A068932(n). - David Wasserman, Mar 08 2002

Row sums of triangle A051031.

More terms from David Wasserman, Mar 08 2002
a(15) and a(16) from Jason Kimberley, Sep 25 2009
Edited by Jason Kimberley, Jan 06 2011 and May 24 2012
a(17)-a(21) from Andrew Howroyd, Mar 08 2020
a(22)-a(24) from Andrew Howroyd, Apr 05 2020

A185325 Number of partitions of n into parts >= 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 13, 15, 18, 21, 26, 30, 36, 42, 50, 58, 70, 80, 95, 110, 129, 150, 176, 202, 236, 272, 317, 364, 423, 484, 560, 643, 740, 847, 975, 1112, 1277, 1456, 1666, 1897, 2168, 2464, 2809, 3189, 3627, 4112, 4673

Offset: 0

Jason Kimberley, Nov 11 2011

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 5 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.
By removing a single part of size 5, an A026798 partition of n becomes an A185325 partition of n - 5. Hence this sequence is essentially the same as A026798.
a(n) = number of partitions of n+4 such that 4*(number of parts) is a part. - Clark Kimberling, Feb 27 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

2-regular simple graphs with girth at least 5: A185115 (connected), A185225 (disconnected), this sequence (not necessarily connected).
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), this sequence (g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).
Not necessarily connected k-regular simple graphs with girth at least 5: A185315 (any k), A185305 (triangle); specified degree k: this sequence (k=2), A185335 (k=3).
Cf. A008484, A008483.

p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A185325 := func;
[A185325(n):n in[0..60]];

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+5): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(1/mul(1-x^(m+5), m = 0..80), x, n+1), x, n), n = 0..70); # G. C. Greubel, Nov 03 2019

Drop[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 4*Length[p]]], {n, 40}], 3]  (* Clark Kimberling, Feb 27 2014 *)
CoefficientList[Series[1/QPochhammer[x^5, x], {x, 0, 70}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+5))) \\ G. C. Greubel, Nov 03 2019

def A185325_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+5)) for m in (0..80)) ).list()
A185325_list(70) # G. C. Greubel, Nov 03 2019

G.f.: Product_{m>=5} 1/(1-x^m).
Given by p(n) -p(n-1) -p(n-2) +2*p(n-5) -p(n-8) -p(n-9) +p(n-10), where p(n) = A000041(n). - Shanzhen Gao, Oct 28 2010 [sign of 10 corrected from + to -, and moved from A026798 to this sequence by Jason Kimberley].
This sequence is the Euler transformation of A185115.
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^4 / (6*sqrt(3)*n^3). - Vaclav Kotesovec, Jun 02 2018
G.f.: exp(Sum_{k>=1} x^(5*k)/(k*(1 - x^k))). - Ilya Gutkovskiy, Aug 21 2018
G.f.: 1 + Sum_{n >= 1} x^(n+4)/Product_{k = 0..n-1} (1 - x^(k+5)). - Peter Bala, Dec 01 2024

A185314 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 2, 7, 3, 14, 6, 44, 37, 350, 1616, 18042, 193919, 2867779, 32674078, 1581632332, 6705889886

Offset: 0

Jason Kimberley, May 23 2012

Jason Kimberley, Not necessarily connected k-regular graphs with girth at least 4
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Regular graphs, of any degree, with girth at least 4: A186724 (connected), A185214 (disconnected), this sequence (not-necessarily connected).
Not necessarily connected k-regular simple graphs with girth at least 4: this sequence (any k), A185304 (triangle); specified degree k: A008484 (k=2), A185334 (k=3), A185344 (k=4), A185354 (k=5), A185364 (k=6).
Not necessarily connected regular simple graphs with girth at least g: A005176 (g=3), this sequence (g=4), A185315 (g=5), A185316 (g=6), A185317 (g=7), A185318 (g=8), A185319 (g=9).

a(n) = A186724(n) + A185214(n).

a(n) is the sum of the n-th row of A185304.

A185335 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 2, 9, 49, 455, 5784, 90940, 1620491, 31478651, 656784488, 14621878339, 345975756388

Offset: 0

Jason Kimberley, Jan 28 2011

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

3-regular simple graphs with girth at least 5: A014372 (connected), A185235 (disconnected), this sequence (not necessarily connected).
Not necessarily connected 3-regular simple graphs with girth *at least* g: A005638 (g=3), A185334 (g=4), this sequence (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).
Not necessarily connected k-regular simple graphs with girth at least 5: A185315 (any k), A185305 (triangle); specified degree k: A185325 (k=2), this sequence (k=3).

A014372 = Cases[Import["https://oeis.org/A014372/b014372.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014372] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

This sequence is the Euler transformation of A014372.

A198314 Number of, not necessarily connected, regular simple graphs on n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 4, 1, 8, 3, 37, 33, 335, 1610, 17985, 193911, 2867313, 32674066, 1581626531, 6705889862

Offset: 0

Jason Kimberley, Dec 12 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected k-regular simple graphs girth exactly 4: this sequence (any k), A185644 (triangle); fixed k: A026797 (k=2), A185134 (k=3), A185144 (k=4).

Not necessarily connected regular simple graphs girth exactly g: A198313 (g=3), this sequence (g=4), A198315 (g=5), A198316 (g=6), A198317 (g=7), A198318 (g=8).

a(n) = A186744(n) + A210714(n).

a(n) = A185314(n) - A185315(n).

a(10) corrected from 9 to 8 by Jason Kimberley, Feb 22 2013

A185316 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 6.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 2, 3, 2, 3, 2, 4, 3, 6, 4, 7, 5, 13, 7, 42, 10, 398, 13, 7592, 18, 181251, 25, 4624534, 33, 122090591, 45, 3328930034, 61, 93990693977, 106

Offset: 0

Jason Kimberley, Dec 12 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Not necessarily connected regular simple graphs with girth at least g: A005176 (g=3), A185314 (g=4), A185315 (g=5), this sequence (g=6), A185317 (g=7), A185318 (g=8), A185319 (g=9).

a(n) = A186726(n) + A185216(n).

A185317 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 7.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 2, 3, 2, 4, 3, 5, 4, 6, 5, 7, 7, 9, 9, 13, 12, 18, 16, 41, 21, 572, 28, 30402, 37, 1782884, 49, 95079141, 64, 4686063195, 84

Offset: 0

Jason Kimberley, Dec 12 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Not necessarily connected regular simple graphs with girth at least g: A005176 (g=3), A185314 (g=4), A185315 (g=5), A185316 (g=6), this sequence (g=7), A185318 (g=8), A185319 (g=9).

a(n) = A186727(n) + A185217(n).

A185318 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 8.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 3, 2, 3, 2, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 9, 8, 11, 11, 14, 14, 19, 18, 23, 24, 30, 31, 41, 40, 61, 52, 217, 67, 4416, 86, 266463, 111, 20807816, 141

Offset: 0

Jason Kimberley, Dec 18 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Not necessarily connected regular simple graphs with girth at least g: A005176 (g=3), A185314 (g=4), A185315 (g=5), A185316 (g=6), A185317 (g=7), this sequence (g=8), A185319 (g=9).

a(n) = A186728(n) + A185218(n).

A185319 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 9.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 8, 10, 10, 13, 13, 16, 17, 20, 21, 26, 27, 32, 35, 41, 44, 52, 56, 65, 72, 82, 90, 104, 114, 130, 144, 163, 180, 205, 226, 255, 283, 336

Offset: 0

Jason Kimberley, Dec 19 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Not necessarily connected regular simple graphs with girth at least g: A005176 (g=3), A185314 (g=4), A185315 (g=5), A185316 (g=6), A185317 (g=7), A185318 (g=8), this sequence (g=9).

a(n) = A186729(n) + A185219(n).

A185305 Triangular array E(n,k) counting not necessarily connected k-regular simple graphs on n vertices with girth at least 5.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 2, 0, 1, 1, 3, 2, 1, 0, 3, 0, 1, 1, 4, 9, 1, 0, 5, 0, 1, 1, 6, 49, 1, 0, 7, 0, 1, 1, 9, 455, 1, 0, 10, 0, 1, 1, 1, 13, 5784, 2, 1, 0, 15, 0, 8, 1, 1, 18, 90940, 131, 1, 0, 21, 0, 3917, 1, 1, 26, 1620491, 123859

Offset: 1

Jason Kimberley, Feb 21 2013

Row sums give A185315.

			01: 1;
02: 1, 1;
03: 1, 0;
04: 1, 1;
05: 1, 0, 1;
06: 1, 1, 1;
07: 1, 0, 1;
08: 1, 1, 1;
09: 1, 0, 1;
10: 1, 1, 2, 1;
11: 1, 0, 2, 0;
12: 1, 1, 3, 2;
13: 1, 0, 3, 0;
14: 1, 1, 4, 9;
15: 1, 0, 5, 0;
16: 1, 1, 6, 49;
17: 1, 0, 7, 0;
18: 1, 1, 9, 455;
19: 1, 0, 10, 0, 1;
20: 1, 1, 13, 5784, 2;
21: 1, 0, 15, 0, 8;
22: 1, 1, 18, 90940, 131;
23: 1, 0, 21, 0, 3917;
24: 1, 1, 26, 1620491, 123859;
25: 1, 0, 30, 0, 4131991;
26: 1, 1, 36, 31478649, 132160608;
27: 1, 0, 42, 0, 4018022149;
28: 1, 1, 50, 656784488, 118369811960;

Jason Kimberley, Table of n, a(i)=E(n,k) for i = 1..108 (n = 1..28)
Jason Kimberley, Not necessarily connected k-regular graphs with girth at least 5
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Not necessarily connected k-regular simple graphs with girth at least 5: A185315 (any k), this sequence (triangle); specified degree k: A185325 (k=2), A185335 (k=3).

E(n,k) = A186715(n,k) + A185205(n,k).

cp's OEIS Frontend

A005176 Number of regular graphs with n unlabeled nodes.

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A185325 Number of partitions of n into parts >= 5.

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A185314 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 4.

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A185335 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 5.

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A198314 Number of, not necessarily connected, regular simple graphs on n vertices with girth exactly 4.

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A185316 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 6.

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A185317 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 7.

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A185318 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 8.

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A185319 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 9.

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A185305 Triangular array E(n,k) counting not necessarily connected k-regular simple graphs on n vertices with girth at least 5.

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