A185163 -id:A185163 - cp's OEIS frontend

A026796 Number of partitions of n in which the least part is 3.

0, 0, 0, 1, 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 17, 21, 25, 33, 39, 49, 60, 73, 88, 110, 130, 158, 191, 230, 273, 331, 391, 468, 556, 660, 779, 927, 1087, 1284, 1510, 1775, 2075, 2438, 2842, 3323, 3872, 4510, 5237, 6095, 7056, 8182, 9465, 10945, 12625

Offset: 0

Clark Kimberling

Let b(k) be the number of partitions of k for which twice the number of ones is the number of parts, k = 0, 1, 2, ... . Then a(n+4) = b(n), n = 0, 1, 2, ... (conjectured). - George Beck, Aug 19 2017

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Essentially the same sequence as A008483.
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), this sequence (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 girth exactly 3: A198313 (any k), A185643 (triangle); fixed k: this sequence (k=2), A185133 (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).

R:=PowerSeriesRing(Integers(), 60); [0,0,0] cat Coefficients(R!( x^3/(&*[1-x^(m+3): m in [0..70]]) )); // G. C. Greubel, Nov 02 2019

seq(coeff(series(x^3/mul(1-x^(m+3), m=0..65), x, n+1), x, n), n = 0 .. 60); # G. C. Greubel, Nov 02 2019

Table[Count[IntegerPartitions[n], p_ /; Min@p==3], {n, 0, 60}] (* George Beck Aug 19 2017 *)
CoefficientList[Series[x^3/QPochhammer[x^3, x], {x,0,60}], x] (* G. C. Greubel, Nov 02 2019 *)

a(n) = numbpart(n-3) - numbpart(n-4) - numbpart(n-5) + numbpart(n-6); \\ Michel Marcus, Aug 20 2014

x='x+O('x^66); Vecrev(Pol(x^3*(1-x)*(1-x^2)/eta(x))) \\ Joerg Arndt, Aug 22 2014

def A026796_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^3/product((1-x^(m+3)) for m in (0..65)) ).list()
A026796_list(60) # G. C. Greubel, Nov 02 2019

G.f.: x^3 / Product_{m>=3} (1 - x^m).
a(n) = p(n-3) - p(n-4) - p(n-5) + p(n-6), where p(n) = A000041(n). - Bob Selcoe, Aug 07 2014
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^2 / (12*sqrt(3)*n^2). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(3*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

More terms from Michel Marcus, Aug 20 2014

a(0) = 0 prepended by Joerg Arndt, Aug 22 2014

A185133 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 1, 1, 4, 15, 71, 428, 3406, 34270, 418621, 5937051, 94782437, 1670327647, 32090011476, 666351752261, 14859579573845

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 3: A198313 (any k), A185643 (triangle); fixed k: A026796 (k=2), this sequence (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).

Not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly g: A185130 (triangle); fixed g: this sequence (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

a(n) = A005638(n) - A185334(n).

a(n) = A006923(n) + A185033(n).

A198313 Number of, not necessarily connected, regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 4, 15, 23, 162, 540, 18958, 389417, 50314520, 2942196930, 1698517018988

Offset: 0

Jason Kimberley, May 24 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 3: this sequence (any k), A185643 (triangle); fixed k: A026796 (k=2), A185133 (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).

Not necessarily connected regular simple graphs girth exactly g: this sequence (g=3), A198314 (g=4), A198315 (g=5), A198316 (g=6), A198317 (g=7), A198318 (g=8).

a(n) = A186743(n) + A210713(n).
a(n) = A005176(n) - A185314(n).
a(n) is the sum of the n-th row of A185643.

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

A185643 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 4, 5, 3, 1, 1, 0, 0, 2, 0, 16, 0, 4, 0, 1, 0, 0, 2, 15, 58, 59, 21, 5, 1, 1, 0, 0, 3, 0, 264, 0, 266, 0, 6, 0, 1, 0, 0, 4, 71, 1535, 7848, 7848, 1547, 94, 9, 1, 1, 0, 0, 5, 0, 10755, 0, 367860, 0, 10786, 0, 10, 0, 1

Offset: 1

Jason Kimberley, Feb 07 2013

			01: 0;
02: 0, 0;
03: 0, 0, 1;
04: 0, 0, 0, 1;
05: 0, 0, 0, 0, 1;
06: 0, 0, 1, 1, 1, 1;
07: 0, 0, 1, 0, 2, 0, 1;
08: 0, 0, 1, 4, 5, 3, 1, 1;
09: 0, 0, 2, 0, 16, 0, 4, 0, 1;
10: 0, 0, 2, 15, 58, 59, 21, 5, 1, 1;
11: 0, 0, 3, 0, 264, 0, 266, 0, 6, 0, 1;
12: 0, 0, 4, 71, 1535, 7848, 7848, 1547, 94, 9, 1, 1;
13: 0, 0, 5, 0, 10755, 0, 367860, 0, 10786, 0, 10, 0, 1;
14: 0, 0, 6, 428, 87973, 3459379, 21609300, 21609300, 3459386, 88193, 540, 13, 1, 1;
15: 0, 0, 9, 0, 803973, 0, 1470293675, 0, 1470293676, 0, 805579, 0, 17, 0, 1;
16: 0, 0, 10, 3406, 8020967, 2585136353, 113314233804, 733351105934, 733351105934, 113314233813, 2585136741, 8037796, 4207, 21, 1, 1;

Jason Kimberley, Table of i, a(i)=E(n,k) for i = 1..136 (n = 1..16)
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 A198313(n).

Not necessarily connected k-regular simple graphs girth exactly 3: A198313 (any k), this sequence (triangle); fixed k: A026796 (k=2), A185133 (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).

E(n,k) = A186733(n,k) + A210703(n,k), noting that A210703 is a tabf.

E(n,k) = A051031(n,k) - A185304(n,k), noting that A185304 is a tabf.

A185153 Number of not necessarily connected 5-regular simple graphs on 2n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 1, 3, 59, 7848, 3459379, 2585136353, 2807104852102

Offset: 0

Jason Kimberley, Mar 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 3: A198313 (any k), A185643 (triangle); fixed k: A026796 (k=2), A185133 (k=3), A185143 (k=4), this sequence (k=5), A185163 (k=6).

a(n) = A165626(n) - A185354(n).

a(n) = A184953(n) + A185053(n).

cp's OEIS Frontend

A026796 Number of partitions of n in which the least part is 3.

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A185133 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth exactly 3.

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A198313 Number of, not necessarily connected, regular simple graphs on n vertices with girth exactly 3.

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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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A185643 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 3.

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A185153 Number of not necessarily connected 5-regular simple graphs on 2n vertices with girth exactly 3.

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