A185329 Number of partitions of n with parts >= 9.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 24, 26, 30, 34, 39, 43, 50, 55, 63, 71, 80, 89, 102, 113, 128, 143, 161, 179, 203, 225, 253, 282, 316, 351, 395, 437, 489, 544, 607, 673, 752, 832, 927, 1028, 1143

Offset: 0

Author

Jason Kimberley, Feb 01 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 9 (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 9, an A026802 partition of n becomes an A185329 partition of n - 9. Hence this sequence is essentially the same as A026802.

In general, if g>=1 and g.f. = Product_{m>=g} 1/(1-x^m), then a(n,g) ~ Pi^(g-1) * (g-1)! * exp(Pi*sqrt(2*n/3)) / (2^((g+3)/2) * 3^(g/2) * n^((g+1)/2)) ~ p(n) * Pi^(g-1) * (g-1)! / (6*n)^((g-1)/2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, Jun 02 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• 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), A185328 (g=8), this sequence (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+9): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

Maple

seq(coeff(series(1/mul(1-x^(m+9), m = 0..80), x, n+1), x, n), n = 0..70); # G. C. Greubel, Nov 03 2019

Mathematica

CoefficientList[Series[x^9/QPochhammer[x^9, 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+9))) \\ G. C. Greubel, Nov 03 2019

Sage

def A185329_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+9)) for m in (0..80)) ).list()
A185329_list(70) # G. C. Greubel, Nov 03 2019

Formula

G.f.: Product_{m>=9} 1/(1-x^m).

a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + p(n-7) + p(n-9) - p(n-11) - 2*p(n-12) - p(n-13) - p(n-15) + p(n-16) + p(n-17) + 2*p(n-18) + p(n-19) + p(n-20) - p(n-21) - p(n-23) - 2*p(n-24) - p(n-25) + p(n-27) + p(n-29) + p(n-31) - p(n-34) - p(n-35) + p(n-36) where p(n)=A000041(n). - Shanzhen Gao

This sequence is the Euler transformation of A185119.

a(n) ~ exp(Pi*sqrt(2*n/3)) * 70*Pi^8 / (9*sqrt(3)*n^5). - Vaclav Kotesovec, Jun 02 2018

G.f.: Sum_{k>=0} x^(9*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020

G.f.: 1 + Sum_{n >= 1} x^(n+8)/Product_{k = 0..n-1} (1 - x^(k+9)). - Peter Bala, Dec 01 2024