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

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

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

Links

• 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

Crossrefs

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).

Programs

Magma

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

Maple

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

Mathematica

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

Sage

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

Formula

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

Extensions

More terms from Michel Marcus, Aug 20 2014

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