A185325 - OEIS

%I #56 Dec 01 2024 10:03:41

%S 1,0,0,0,0,1,1,1,1,1,2,2,3,3,4,5,6,7,9,10,13,15,18,21,26,30,36,42,50,

%T 58,70,80,95,110,129,150,176,202,236,272,317,364,423,484,560,643,740,

%U 847,975,1112,1277,1456,1666,1897,2168,2464,2809,3189,3627,4112,4673

%N Number of partitions of n into parts >= 5.

%C a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 5 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.

%C By removing a single part of size 5, an A026798 partition of n becomes an A185325 partition of n - 5. Hence this sequence is essentially the same as A026798.

%C a(n) = number of partitions of n+4 such that 4*(number of parts) is a part. - _Clark Kimberling_, Feb 27 2014

%H G. C. Greubel, <a href="/A185325/b185325.txt">Table of n, a(n) for n = 0..1000</a>

%H Kevin Beanland and Hung Viet Chu, <a href="https://arxiv.org/abs/2311.01926">On Schreier-type Sets, Partitions, and Compositions</a>, arXiv:2311.01926 [math.CO], 2023.

%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/E_k-reg_girth_ge_g_index">Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g</a>

%F G.f.: Product_{m>=5} 1/(1-x^m).

%F Given by p(n) -p(n-1) -p(n-2) +2*p(n-5) -p(n-8) -p(n-9) +p(n-10), where p(n) = A000041(n). - _Shanzhen Gao_, Oct 28 2010 [sign of 10 corrected from + to -, and moved from A026798 to this sequence by _Jason Kimberley_].

%F This sequence is the Euler transformation of A185115.

%F a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^4 / (6*sqrt(3)*n^3). - _Vaclav Kotesovec_, Jun 02 2018

%F G.f.: exp(Sum_{k>=1} x^(5*k)/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Aug 21 2018

%F G.f.: 1 + Sum_{n >= 1} x^(n+4)/Product_{k = 0..n-1} (1 - x^(k+5)). - _Peter Bala_, Dec 01 2024

%p seq(coeff(series(1/mul(1-x^(m+5), m = 0..80), x, n+1), x, n), n = 0..70); # _G. C. Greubel_, Nov 03 2019

%t Drop[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 4*Length[p]]], {n, 40}], 3]  (* _Clark Kimberling_, Feb 27 2014 *)

%t CoefficientList[Series[1/QPochhammer[x^5, x], {x, 0, 70}], x] (* _G. C. Greubel_, Nov 03 2019 *)

%o (Magma)

%o p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;

%o A185325 := func<n | p(n)-p(n-1)-p(n-2)+2*p(n-5)-p(n-8)-p(n-9)+p(n-10)>;

%o [A185325(n):n in[0..60]];

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+5): m in [0..80]]) )); // _G. C. Greubel_, Nov 03 2019

%o (PARI) my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+5))) \\ _G. C. Greubel_, Nov 03 2019

%o (Sage)

%o def A185325_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( 1/product((1-x^(m+5)) for m in (0..80)) ).list()

%o A185325_list(70) # _G. C. Greubel_, Nov 03 2019

%Y 2-regular simple graphs with girth at least 5: A185115 (connected), A185225 (disconnected), this sequence (not necessarily connected).

%Y 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), this sequence (g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).

%Y 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).

%Y Not necessarily connected k-regular simple graphs with girth at least 5: A185315 (any k), A185305 (triangle); specified degree k: this sequence (k=2), A185335 (k=3).

%Y Cf. A008484, A008483.

%K nonn,easy

%O 0,11

%A _Jason Kimberley_, Nov 11 2011