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%I #24 Sep 08 2022 08:44:49
%S 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,
%T 4,5,5,6,6,8,8,10,11,13,14,17,18,21,23,27,29,34,37,43,47,54,59,68,74,
%U 85,93,106,116,132,145,164,180,203,223,252,276,310,341,382,420,470,516,576,633,706,775,863
%N Number of partitions of n in which the least part is 10.
%C In general, if g>=1 and g.f. = x^g * 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="/A026803/b026803.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^10 * Product_{m>=10} 1/(1-x^m).
%F a(n) ~ exp(Pi*sqrt(2*n/3)) * 35*sqrt(2)*Pi^9 / (3*n^(11/2)). - _Vaclav Kotesovec_, Jun 02 2018
%F G.f.: Sum_{k>=1} x^(10*k) / Product_{j=1..k-1} (1 - x^j). - _Ilya Gutkovskiy_, Nov 25 2020
%p seq(coeff(series(x^10/mul(1-x^(m+10), m = 0..85), x, n+1), x, n), n = 1..80); # _G. C. Greubel_, Nov 03 2019
%t Rest@CoefficientList[Series[x^10/QPochhammer[x^10, x], {x,0,80}], x] (* _G. C. Greubel_, Nov 03 2019 *)
%o (PARI) my(x='x+O('x^80)); concat(vector(9), Vec(x^10/prod(m=0,85, 1-x^(m+10)))) \\ _G. C. Greubel_, Nov 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 80); [0,0,0,0,0,0,0,0,0] cat Coefficients(R!( x^10/(&*[1-x^(m+10): m in [0..85]]) )); // _G. C. Greubel_, Nov 03 2019
%o (Sage)
%o def A026803_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x^10/product((1-x^(m+10)) for m in (0..85)) ).list()
%o a=A026803_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), A026801 (g=8), A026802 (g=9), A026803 (g=10).
%K nonn,easy
%O 1,30
%A _Clark Kimberling_
%E More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001
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