A008484 Number of partitions of n into parts >= 4.

1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 11, 12, 16, 18, 24, 27, 34, 39, 50, 57, 70, 81, 100, 115, 140, 161, 195, 225, 269, 311, 371, 427, 505, 583, 688, 791, 928, 1067, 1248, 1434, 1668, 1914, 2223, 2546, 2945, 3370, 3889, 4443, 5113, 5834, 6698

Offset: 0

Author

T. Forbes (anthony.d.forbes(AT)googlemail.com)

Keywords

• nonn
• easy

Comments

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 4 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Jan 2011 and Feb 2012

By removing a single part of size 4, an A026797 partition of n becomes an A008484 partition of n - 4. Hence this sequence is essentially the same as A026797. - Jason Kimberley, Feb 2012

Number of partitions of n+3 such that 3*(number of parts) is a part. - Clark Kimberling, Feb 27 2014

Let c(n) be the number of partitions of n such that both (number of parts) and 2*(number of parts) are parts; then c(n) = a(n-6) for n >= 6 and c(n) = 0 for n < 6. - Clark Kimberling, Mar 01 2014

a(n) is also the number of partitions of n for which three times the number of ones is twice the number of parts (conjectured). - George Beck, Aug 19 2017

Proof: Above definition is equivalent to 2 out of 3 parts being equal to 1. Arrange in triples 1, 1, >= 2, etc. Sum of each triple corresponds to sequence definition. - Martin Fuller, Aug 21 2023

Links

• Giovanni Resta, 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, Not necessarily connected k-regular graphs with girth at least 4
• Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Crossrefs

2-regular graphs with girth at least 4: A185114 (connected), A185224 (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), this sequence (g=4), A185325(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 4: A185314 (any k), A185304 (triangle); specified degree k: this sequence (k=2), A185334 (k=3), A185344 (k=4), A185354 (k=5), A185364 (k=6).

Programs

Magma

a:= func< n | NumberOfPartitions(n)-NumberOfPartitions(n-1)-NumberOfPartitions(n-2)+ NumberOfPartitions(n-4)+NumberOfPartitions(n-5)- NumberOfPartitions(n-6) >; [1,0,0,0,1,1,1] cat [ a(n) : n in [7..60]]; // Vincenzo Librandi, Aug 20 2017

Magma

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/(&*[1-x^(m+4): m in [0..70]]) )); // G. C. Greubel, Nov 03 2019

Maple

series(1/product((1-x^i),i=4..65),x,60); # end of program
ZL := [ B,{B=Set(Set(Z, card>=4))}, unlabeled ]: seq(combstruct[count](ZL, size=n), n=0..60); # Zerinvary Lajos, Mar 13 2007

Mathematica

f[1, 1]=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, 4], {n, 60}] (* end of program *)
Drop[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 3*Length[p]]], {n, 60}],2]  (* Clark Kimberling, Feb 27 2014 *)
Table[Count[IntegerPartitions[n],
  p_ /; 3 Count[p, 1] == 2 Length[p]], {n, 0, 60}] (* George Beck Aug 19 2017 *)
CoefficientList[Series[1/QPochhammer[x^4, x], {x,0,60}], x] (* G. C. Greubel, Nov 03 2019 *)

PARI

my(x='x+O('x^60)); Vec(1/prod(m=0,70, 1-x^(m+4))) \\ G. C. Greubel, Nov 03 2019

Sage

def A008484_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+4)) for m in (0..70)) ).list()
A008484_list(60) # G. C. Greubel, Nov 03 2019

Formula

G.f.: 1 / Product_{m>=4} (1 - x^m).

Euler transformation of A185114. - Jason Kimberley, Jan 30 2011

Given by p(n) - p(n-1) - p(n-2) + p(n-4) + p(n-5) - p(n-6) where p(n) = A000041(n). Generally, 1/Product_{i>=K} (1 - x^i) is given by p({A}), where {A} is defined over the coefficients of Product_{i=1..K-1} (1 - x^i). In this case, K=4, so (1-x)(1-x^2)(1-x^3) = 1 - x - x^2 + x^4 + x^5 - x^6, defining {A} as above. G.f.: 1 + Sum_{i>=1} (x^4i)/Product_{j=1..i}(1 - x^j). - Jon Perry, Jul 04 2004

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^3 / (12*sqrt(2)*n^(5/2)). - Vaclav Kotesovec, Jun 02 2018

G.f.: exp(Sum_{k>=1} x^(4*k)/(k*(1 - x^k))). - Ilya Gutkovskiy, Aug 21 2018

G.f.: 1 + Sum_{n >= 1} x^(n+3)/Product_{k = 0..n-1} (1 - x^(k+4)). - Peter Bala, Dec 01 2024