0,10

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

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

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 *)

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

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

(Magma) R<x>:=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

(SageMath)

def A026796_list(prec):

P.<x> = 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

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).

Sequence in context: A027195 A008483 A281356 * A008925 A266749 A308283

Adjacent sequences: A026793 A026794 A026795 * A026797 A026798 A026799

nonn,easy

More terms from Michel Marcus, Aug 20 2014

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

approved