A185326 Number of partitions of n into parts >= 6.

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 16, 17, 21, 24, 29, 32, 40, 44, 53, 60, 71, 80, 96, 107, 126, 143, 167, 188, 221, 248, 288, 326, 376, 424, 491, 552, 634, 716, 819, 922, 1056, 1187, 1353, 1523, 1730, 1944, 2209, 2478, 2806, 3151

Offset: 0

Author

Jason Kimberley, Jan 30 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 6 (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 6, an A026799 partition of n becomes an A185326 partition of n - 6. Hence this sequence is essentially the same as A026799.

Links

• Jason Kimberley, Table of n, a(n) for n = 0..998
• 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 6: A185116 (connected), A185226 (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), A185325(g=5), this sequence (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).

Programs

Magma

A185326 := func;

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+6): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

Maple

seq(coeff(series(1/mul(1-x^(m+6), m = 0..80), x, n+1), x, n), n = 0..70); # G. C. Greubel, Nov 03 2019

Mathematica

CoefficientList[Series[1/QPochhammer[x^6, 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+6))) \\ G. C. Greubel, Nov 03 2019

Sage

def A185326_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+6)) for m in (0..80)) ).list()
A185326_list(70) # G. C. Greubel, Nov 03 2019

Formula

G.f.: Product_{m>=6} 1/(1-x^m).

a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + p(n-6) + p(n-7) - p(n-8) - p(n-9) - p(n-10) + p(n-13) + p(n-14) - p(n-15) where p(n) = A000041(n).

a(n) = A185226(n) + A185116(n).

This sequence is the Euler transformation of A185116.

a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^5 / (18*sqrt(2)*n^(7/2)). - Vaclav Kotesovec, Jun 02 2018

G.f.: Sum_{k>=0} x^(6*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020

G.f.: 1 + Sum_{n >= 1} x^(n+5)/Product_{k = 0..n-1} (1 - x^(k+6)). - Peter Bala, Dec 01 2024