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%I #29 Sep 08 2022 08:45:55
%S 1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,6,7,8,9,11,12,14,
%T 16,18,20,24,26,30,34,39,43,50,55,63,71,80,89,102,113,128,143,161,179,
%U 203,225,253,282,316,351,395,437,489,544,607,673,752,832,927,1028,1143
%N Number of partitions of n with parts >= 9.
%C a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 9 (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 9, an A026802 partition of n becomes an A185329 partition of n - 9. Hence this sequence is essentially the same as A026802.
%C In general, if g>=1 and g.f. = Product_{m>=g} 1/(1-x^m), then a(n,g) ~ Pi^(g-1) * (g-1)! * exp(Pi*sqrt(2*n/3)) / (2^((g+3)/2) * 3^(g/2) * n^((g+1)/2)) ~ p(n) * Pi^(g-1) * (g-1)! / (6*n)^((g-1)/2), where p(n) is the partition function A000041(n). - _Vaclav Kotesovec_, Jun 02 2018
%H G. C. Greubel, <a href="/A185329/b185329.txt">Table of n, a(n) for n = 0..1000</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>=9} 1/(1-x^m).
%F a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + p(n-7) + p(n-9) - p(n-11) - 2*p(n-12) - p(n-13) - p(n-15) + p(n-16) + p(n-17) + 2*p(n-18) + p(n-19) + p(n-20) - p(n-21) - p(n-23) - 2*p(n-24) - p(n-25) + p(n-27) + p(n-29) + p(n-31) - p(n-34) - p(n-35) + p(n-36) where p(n)=A000041(n). - _Shanzhen Gao_
%F This sequence is the Euler transformation of A185119.
%F a(n) ~ exp(Pi*sqrt(2*n/3)) * 70*Pi^8 / (9*sqrt(3)*n^5). - _Vaclav Kotesovec_, Jun 02 2018
%F G.f.: Sum_{k>=0} x^(9*k) / Product_{j=1..k} (1 - x^j). - _Ilya Gutkovskiy_, Nov 28 2020
%p seq(coeff(series(1/mul(1-x^(m+9), m = 0..80), x, n+1), x, n), n = 0..70); # _G. C. Greubel_, Nov 03 2019
%t CoefficientList[Series[x^9/QPochhammer[x^9, 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+9))) \\ _G. C. Greubel_, Nov 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+9): m in [0..80]]) )); // _G. C. Greubel_, Nov 03 2019
%o (Sage)
%o def A185329_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 1/product((1-x^(m+9)) for m in (0..80)) ).list()
%o A185329_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), A185328 (g=8), this sequence (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 0,19
%A _Jason Kimberley_, Feb 01 2012
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