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

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

Author

Jason Kimberley, Nov 11 2011

Keywords

• nonn
• easy

Comments

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

Links

• 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

Crossrefs

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.

Programs

Magma

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

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Sage

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

Formula

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