A185144 -id:A185144 - cp's OEIS frontend

A026797 Number of partitions of n in which the least part is 4.

0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 11, 12, 16, 18, 24, 27, 34, 39, 50, 57, 70, 81, 100, 115, 140, 161, 195, 225, 269, 311, 371, 427, 505, 583, 688, 791, 928, 1067, 1248, 1434, 1668, 1914, 2223, 2546, 2945, 3370, 3889

Offset: 1

Clark Kimberling

a(n) is also the number of, not necessarily connected, 2-regular simple graphs girth exactly 4. - Jason Kimberley, Feb 22 2013

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Essentially the same as A008484.
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), A185325(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 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), this sequence (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 girth exactly 4: A198314 (any k), A185644 (triangle); fixed k: this sequence (k=2), A185134 (k=3), A185144 (k=4).

R:=PowerSeriesRing(Integers(), 60); [0,0,0] cat Coefficients(R!( x^4/(&*[1-x^(m+4): m in [0..70]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(x^4/mul(1-x^(m+4), m=0..65), x, n+1), x, n), n = 1..60); # G. C. Greubel, Nov 03 2019

Table[Count[IntegerPartitions[n],?(Min[#]==4&)],{n,60}] (* _Harvey P. Dale, May 13 2012 *)
Rest@CoefficientList[Series[x^4/QPochhammer[x^4, x], {x,0,60}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^60)); concat([0,0,0], Vec(x^4/prod(m=0,70, 1-x^(m+4)))) \\ G. C. Greubel, Nov 03 2019

def A026797_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^4/product((1-x^(m+4)) for m in (0..60)) ).list()
a=A026797_list(60); a[1:] # G. C. Greubel, Nov 03 2019

G.f.: x^4 * Product_{m>=4} 1/(1-x^m).
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^3 / (12*sqrt(2)*n^(5/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(4*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

A184944 Number of connected 4-regular simple graphs on n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 12, 31, 220, 1606, 16828, 193900, 2452818, 32670329, 456028472, 6636066091, 100135577616, 1582718909051

Offset: 0

Jason Kimberley, Jan 26 2011

			a(0)=0 because even though the null graph (on zero vertices) is vacuously 4-regular and connected, since it is acyclic, it has infinite girth.
The a(8)=1 graph is the complete bipartite graph K_{4,4}.

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

4-regular simple graphs with girth exactly 4: this sequence (connected), A185044 (disconnected), A185144 (not necessarily connected).
Connected k-regular simple graphs with girth exactly 4: A006924 (k=3), this sequence (k=4), A184954 (k=5), A184964 (k=6), A184974 (k=7).
Connected 4-regular simple graphs with girth at least g: A006820 (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: A184943 (g=3), this sequence (g=4), A184945 (g=5).

a(n) = A033886(n) - A058343(n).

a(23) was appended by the author once A033886(23) was known, Nov 03 2011

A185134 Number of, not necessarily connected, 3-regular simple graphs on 2n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 21, 103, 752, 7385, 91939, 1345933, 22170664, 401399440, 7887389438, 166897766824, 3781593764772

Offset: 0

Jason Kimberley, Mar 21 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: A198314 (any k), A185644 (triangle); fixed k: A026797 (k=2), this sequence (k=3), A185144 (k=4).

Not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g: A185130 (triangle); fixed g: A185133 (g=3), this sequence (g=4), A185135 (g=5), A185136 (g=6).

a(n) = A185334(n) - A185335(n).

a(n) = A006924(n) + A185034(n).

A185143 Number of not necessarily connected 4-regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 5, 16, 58, 264, 1535, 10755, 87973, 803973, 8020967, 86029760, 983431053, 11913921910, 152352965278, 2050065073002, 28951233955602, 428086557232387

Offset: 0

Jason Kimberley, Mar 12 2012

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

4-regular simple graphs with girth exactly 3: A184943 (connected), A185043 (disconnected), this sequence (not necessarily connected).
Not necessarily connected k-regular simple graphs girth exactly 3: A198313 (any k), A185643 (triangle); fixed k: A026796 (k=2), A185133 (k=3), this sequence (k=4), A185153 (k=5), A185163 (k=6).
Not necessarily connected 4-regular simple graphs with girth exactly g: A185140 (triangle); fixed g: this sequence (g=3), A185144 (g=4).

a(n) = A033301(n) - A185344(n).

a(n) = A184943(n) + A185043(n).

a(22) corrected and a(23) appended, due to the correction and extension of A033301 by Andrew Howroyd, from Jason Kimberley, Mar 14 2020

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

A185644 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 5, 2, 1, 0, 0, 1, 0, 2, 0, 0, 0, 2, 21, 12, 1, 1, 0, 0, 2, 0, 31, 0, 0, 0, 0, 3, 103, 220, 7, 1, 1, 0, 0, 3, 0, 1606, 0, 1, 0, 0, 0, 5, 752, 16829, 388, 9, 1, 1, 0, 0, 5, 0, 193900, 0, 6, 0, 0, 0

Offset: 1

Jason Kimberley, Feb 22 2013

In the n-th row 0 <= 2k <= n.

			01: 0;
02: 0, 0;
03: 0, 0;
04: 0, 0, 1;
05: 0, 0, 0;
06: 0, 0, 0, 1;
07: 0, 0, 0, 0;
08: 0, 0, 1, 2, 1;
09: 0, 0, 1, 0, 0;
10: 0, 0, 0, 5, 2, 1;
11: 0, 0, 1, 0, 2, 0;
12: 0, 0, 2, 21, 12, 1, 1;
13: 0, 0, 2, 0, 31, 0, 0;
14: 0, 0, 3, 103, 220, 7, 1, 1;
15: 0, 0, 3, 0, 1606, 0, 1, 0;
16: 0, 0, 5, 752, 16829, 388, 9, 1, 1;
17: 0, 0, 5, 0, 193900, 0, 6, 0, 0;
18: 0, 0, 7, 7385, 2452820, 406824, 267, 8, 1, 1;
19: 0, 0, 8, 0, 32670331, 0, 3727, 0, 0, 0;
20: 0, 0, 11, 91939, 456028487, 1125022326, 483012, 741, 13, 1, 1;
21: 0, 0, 12, 0, 6636066126, 0, 69823723, 0, 1, 0, 0;
22: 0, 0, 16, 1345933, 100135577863, 3813549359275, 14836130862, 2887493, ?, 14, 1;

Jason Kimberley, Table of n, a(i)=E(n,k) for i = 1..139 (n = 1..22)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

The sum of the n-th row of this sequence is A198314(n).

Not necessarily connected k-regular simple graphs girth exactly 4: A198314 (any k), this sequence (triangle); fixed k: A026797 (k=2), A185134 (k=3), A185144 (k=4).

E(n,k) = A186734(n,k) + A210704(n,k), noting the differing row lengths.

E(n,k) = A185304(n,k) - A185305(n,k), noting the differing row lengths.

A185044 Number of disconnected 4-regular simple graphs on n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 15, 35, 247, 1692, 17409, 197924, 2492824, 33117880, 461597957, 6709514218, 101153412903, 1597440868898

Offset: 0

Jason Kimberley, Nov 04 2011

Only one component need have girth exactly four; the other components need only have girth at least four.

First differs from A185244 at n = 38, the smallest n where A185245 is nonzero.

Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth exactly g

Disconnected 4-regular simple graphs with girth exactly g: A185043 (g=3), this sequence (g=4).

Disconnected k-regular simple graphs with girth exactly 4: A185034 (k=3), this sequence (k=4).

a(n) = A185244(n) - A185245(n).

a(n) = A185144(n) - A184944(n).

a(31) corrected by the author, propagated from A185244, Jan 05 2013

A185140 Irregular triangle E(n,g) counting not necessarily connected 4-regular simple graphs on n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 2, 5, 1, 16, 0, 58, 2, 264, 2, 1535, 12, 10755, 31, 87973, 220, 803973, 1606, 8020967, 16829, 86029760, 193900, 983431053, 2452820, 11913921910, 32670331, 1, 152352965278, 456028487, 2, 2050065073002, 6636066126, 8, 28466234288520, 100135577863, 131, 8020967, 16829

Offset: 5

Jason Kimberley, Jan 06 2013

The first column is for girth at least 3. The column for girth g commences when n reaches A037233(g).

			05: 1;
06: 1;
07: 2;
08: 5, 1;
09: 16, 0;
10: 58, 2;
11: 264, 2;
12: 1535, 12;
13: 10755, 31;
14: 87973, 220;
15: 803973, 1606;
16: 8020967, 16829;
17: 86029760, 193900;
18: 983431053, 2452820;
19: 11913921910, 32670331, 1;
20: 152352965278, 456028487, 2;
21: 2050065073002, 6636066126, 8;
22: 28466234288520, 100135577863, 131;

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

Initial columns of this triangle: A185143 (g=3), A185144 (g=4).

The n-th row is the sequence of differences of the n-th row of A185340:
E(n,g) = A185340(n,g) - A185340(n,g+1), once we have appended 0 to each row of A185340.
Hence the sum of the n-th row is A185340(n,3) = A033301(n).

cp's OEIS Frontend

A026797 Number of partitions of n in which the least part is 4.

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A184944 Number of connected 4-regular simple graphs on n vertices with girth exactly 4.

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A185134 Number of, not necessarily connected, 3-regular simple graphs on 2n vertices with girth exactly 4.

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A185143 Number of not necessarily connected 4-regular simple graphs on n vertices with girth exactly 3.

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

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A185644 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 4.

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A185044 Number of disconnected 4-regular simple graphs on n vertices with girth exactly 4.

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A185140 Irregular triangle E(n,g) counting not necessarily connected 4-regular simple graphs on n vertices with girth exactly g.

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