A185328 Number of partitions of n with parts >= 8.

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 21, 23, 27, 30, 36, 39, 46, 51, 60, 66, 77, 85, 99, 110, 126, 140, 162, 179, 205, 228, 260, 289, 329, 365, 415, 461, 521, 579, 655, 726, 818, 909, 1022, 1134, 1273, 1411

Offset: 0

Author

Jason Kimberley, Jan 31 2012

Keywords

• nonn
• easy

Comments

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 8 (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 8, an A026801 partition of n becomes an A185328 partition of n - 8. Hence this sequence is essentially the same as A026801.

Links

• Robert Israel, Table of n, a(n) for n = 0..2000
• Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Crossrefs

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), this sequence (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).

Programs

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+8): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

Maple

N:= 100: # for a(0)..a(N)
g:= mul(1/(1-x^m),m=8..N):
S:= series(g,x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Dec 19 2017

Mathematica

CoefficientList[Series[1/QPochhammer[x^8, x], {x,0,75}], x] (* G. C. Greubel, Nov 03 2019 *)

PARI

my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+8))) \\ G. C. Greubel, Nov 03 2019

Sage

def A185328_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+8)) for m in (0..80)) ).list()
A185328_list(70) # G. C. Greubel, Nov 03 2019

Formula

G.f.: Product_{m>=8} 1/(1-x^m).

a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + p(n-7) + p(n-8) - p(n-10) - p(n-11) - 2*p(n-12) + 2*p(n-16) + p(n-17) + p(n-18) - p(n-20) - p(n-21) - p(n-23) + p(n-26) + p(n-27) - p(n-28) where p(n)=A000041(n). - Shanzhen Gao

This sequence is the Euler transformation of A185118.

a(n) ~ exp(Pi*sqrt(2*n/3)) * 35*Pi^7 / (18*sqrt(2)*n^(9/2)). - Vaclav Kotesovec, Jun 02 2018

G.f.: Sum_{k>=0} x^(8*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020

G.f.: 1 + Sum_{n >= 1} x^(n+7)/Product_{k = 0..n-1} (1 - x^(k+8)). - Peter Bala, Dec 01 2024