A185327 Number of partitions of n into parts >= 7.

1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 18, 20, 24, 27, 32, 36, 42, 48, 56, 63, 73, 83, 96, 108, 125, 141, 162, 183, 209, 236, 270, 304, 346, 390, 443, 498, 565, 635, 719, 807, 911, 1022, 1153, 1291, 1453, 1628, 1829, 2045

Offset: 0

Author

Jason Kimberley, Feb 03 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 7 (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 7, an A026800 partition of n becomes an A185327 partition of n - 7. Hence this sequence is essentially the same as A026800.

Examples

			The  a(0)=1 empty partition vacuously has each part >= 7.
The  a(7)=1 partition is 7.
The  a(8)=1 partition is 8.
............................
The a(13)=1 partition is 13.
The a(14)=2 partitions are 7+7 and 14.
		

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

2-regular simple graphs with girth at least 7: A185117 (connected), A185227 (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), A185326 (g=6), this sequence (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

p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A185327 := func< n | p(n)-p(n-1)-p(n-2)+p(n-5)+2*p(n-7)-p(n-9)-p(n-10)- p(n-11)-p(n-12)+2*p(n-14)+p(n-16)-p(n-19)-p(n-20)+p(n-21) >;

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+7): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

Maple

seq(coeff(series(1/mul(1-x^(m+7), m = 0..80), x, n+1), x, n), n = 0..70); # G. C. Greubel, Nov 03 2019

Mathematica

f[1, 1] = f[0, k_] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 7], {n, 0, 65}] (* Robert G. Wilson v, Jan 31 2011 *) (* moved from A026800 by Jason Kimberley, Feb 03 2011 *)
Join[{1},Table[Count[IntegerPartitions[n],?(Min[#]>=7&)],{n,0,70}]] (* _Harvey P. Dale, Oct 16 2011 *)
CoefficientList[Series[1/QPochhammer[x^7, 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+7))) \\ G. C. Greubel, Nov 03 2019

Sage

def A185327_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+7)) for m in (0..80)) ).list()
A185327_list(70) # G. C. Greubel, Nov 03 2019

Formula

G.f.: Product_{m>=7} 1/(1-x^m).

a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + 2*p(n-7) - p(n-9) - p(n-10) - p(n-11) - p(n-12) + 2*p(n-14) + p(n-16) - p(n-19) - p(n-20) + p(n-21) where p(n)=A000041(n). - Shanzhen Gao, Oct 28 2010 [moved/copied from A026800 by Jason Kimberley, Feb 03 2011]

This sequence is the Euler transformation of A185117.

a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^6 / (6*sqrt(3)*n^4). - Vaclav Kotesovec, Jun 02 2018

G.f.: Sum_{k>=0} x^(7*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020

G.f.: 1 + Sum_{n >= 1} x^(n+6)/Product_{k = 0..n-1} (1 - x^(k+7)). - Peter Bala, Dec 01 2024