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%I #27 Sep 08 2022 08:44:49
%S 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,
%T 6,7,8,9,11,12,14,16,18,20,24,26,30,34,39,43,50,55,63,71,80,89,102,
%U 113,128,143,161,179,203,225,253,282,316,351,395,437,489,544,607,673,752,832,927,1028,1143
%N Number of partitions of n in which the least part is 9.
%H G. C. Greubel, <a href="/A026802/b026802.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^9 * Product_{m>=9} 1/(1-x^m).
%F a(n+9) = 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_, Oct 28 2010
%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>=1} x^(9*k) / Product_{j=1..k-1} (1 - x^j). - _Ilya Gutkovskiy_, Nov 25 2020
%p seq(coeff(series(x^9/mul(1-x^(m+9), m = 0..85), x, n+1), x, n), n = 1..80); # _G. C. Greubel_, Nov 03 2019
%t Table[Count[IntegerPartitions[n],_?(Min[#]==9&)],{n,80}] (* _Harvey P. Dale_, May 09 2013 *)
%t Rest@CoefficientList[Series[x^9/QPochhammer[x^9, x], {x,0,80}], x] (* _G. C. Greubel_, Nov 03 2019 *)
%o (PARI) my(x='x+O('x^70)); concat(vector(8), Vec(x^9/prod(m=0,85, 1-x^(m+9)))) \\ _G. C. Greubel_, Nov 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 80); [0,0,0,0,0,0,0,0] cat Coefficients(R!( x^9/(&*[1-x^(m+9): m in [0..85]]) )); // _G. C. Greubel_, Nov 03 2019
%o (Sage)
%o def A026802_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x^9/product((1-x^(m+9)) for m in (0..85)) ).list()
%o a=A026802_list(81); 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), A026801 (g=8), this sequence (g=9), A026803 (g=10).
%K nonn,easy
%O 1,27
%A _Clark Kimberling_
%E More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001
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