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%I #25 Sep 08 2022 08:44:49
%S 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,7,7,9,
%T 10,12,13,16,17,21,23,27,30,36,39,46,51,60,66,77,85,99,110,126,140,
%U 162,179,205,228,260,289,329,365,415,461,521,579,655,726,818,909,1022,1134,1273,1411
%N Number of partitions of n in which the least part is 8.
%H G. C. Greubel, <a href="/A026801/b026801.txt">Table of n, a(n) for n = 1..1000</a>
%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/E_k-reg_girth_eq_g_index">Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g</a>
%F G.f.: x^8 * Product_{m>=8} 1/(1-x^m).
%F a(n+8) = 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_, Oct 28 2010
%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>=1} x^(8*k) / Product_{j=1..k-1} (1 - x^j). - _Ilya Gutkovskiy_, Nov 25 2020
%p seq(coeff(series(x^8/mul(1-x^(m+8), m = 0..80), x, n+1), x, n), n = 1..70); # _G. C. Greubel_, Nov 03 2019
%t Rest@CoefficientList[Series[x^8/QPochhammer[x^8, x], {x, 0, 75}], x] (* _G. C. Greubel_, Nov 03 2019 *)
%o (PARI) my(x='x+O('x^70)); concat(vector(7), Vec(x^8/prod(m=0,80, 1-x^(m+8)))) \\ _G. C. Greubel_, Nov 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); [0,0,0,0,0,0,0] cat Coefficients(R!( x^8/(&*[1-x^(m+8): m in [0..80]]) )); // _G. C. Greubel_, Nov 03 2019
%o (Sage)
%o def A026801_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x^8/product((1-x^(m+8)) for m in (0..80)) ).list()
%o a=A026801_list(71); a[1:] # _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), 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 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), this sequence (g=8), A026802 (g=9), A026803 (g=10).
%K nonn,easy
%O 1,24
%A _Clark Kimberling_
%E More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001
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