A185328 - OEIS

%I #37 Dec 03 2024 12:32:41

%S 1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,7,7,9,10,12,13,16,17,

%T 21,23,27,30,36,39,46,51,60,66,77,85,99,110,126,140,162,179,205,228,

%U 260,289,329,365,415,461,521,579,655,726,818,909,1022,1134,1273,1411

%N Number of partitions of n with parts >= 8.

%C a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 8 (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 8, an A026801 partition of n becomes an A185328 partition of n - 8. Hence this sequence is essentially the same as A026801.

%H Robert Israel, <a href="/A185328/b185328.txt">Table of n, a(n) for n = 0..2000</a>

%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>=8} 1/(1-x^m).

%F a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + p(n-7) + p(n-8) - p(n-10) - p(n-11) - 2*p(n-12) + 2*p(n-16) + p(n-17) + p(n-18) - p(n-20) - p(n-21) - p(n-23) + p(n-26) + p(n-27) - p(n-28) where p(n)=A000041(n). - _Shanzhen Gao_

%F This sequence is the Euler transformation of A185118.

%F a(n) ~ exp(Pi*sqrt(2*n/3)) * 35*Pi^7 / (18*sqrt(2)*n^(9/2)). - _Vaclav Kotesovec_, Jun 02 2018

%F G.f.: Sum_{k>=0} x^(8*k) / Product_{j=1..k} (1 - x^j). - _Ilya Gutkovskiy_, Nov 28 2020

%F G.f.: 1 + Sum_{n >= 1} x^(n+7)/Product_{k = 0..n-1} (1 - x^(k+8)). - _Peter Bala_, Dec 01 2024

%p N:= 100: # for a(0)..a(N)

%p g:= mul(1/(1-x^m),m=8..N):

%p S:= series(g,x,N+1):

%p seq(coeff(S,x,n),n=0..N); # _Robert Israel_, Dec 19 2017

%t CoefficientList[Series[1/QPochhammer[x^8, x], {x,0,75}], x] (* _G. C. Greubel_, Nov 03 2019 *)

%o (PARI) my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+8))) \\ _G. C. Greubel_, Nov 03 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+8): m in [0..80]]) )); // _G. C. Greubel_, Nov 03 2019

%o (Sage)

%o def A185328_list(prec):

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

%o     return P( 1/product((1-x^(m+8)) for m in (0..80)) ).list()

%o A185328_list(70) # _G. C. Greubel_, Nov 03 2019

%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), A185325(g=5), A185326 (g=6), A185327 (g=7), this sequence (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).

%K nonn,easy

%O 0,17

%A _Jason Kimberley_, Jan 31 2012