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%I #36 Sep 08 2022 08:44:49
%S 0,0,0,1,0,0,0,1,1,1,1,2,2,3,3,5,5,7,8,11,12,16,18,24,27,34,39,50,57,
%T 70,81,100,115,140,161,195,225,269,311,371,427,505,583,688,791,928,
%U 1067,1248,1434,1668,1914,2223,2546,2945,3370,3889
%N Number of partitions of n in which the least part is 4.
%C a(n) is also the number of, not necessarily connected, 2-regular simple graphs girth exactly 4. - _Jason Kimberley_, Feb 22 2013
%H G. C. Greubel, <a href="/A026797/b026797.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^4 * Product_{m>=4} 1/(1-x^m).
%F a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^3 / (12*sqrt(2)*n^(5/2)). - _Vaclav Kotesovec_, Jun 02 2018
%F G.f.: Sum_{k>=1} x^(4*k) / Product_{j=1..k-1} (1 - x^j). - _Ilya Gutkovskiy_, Nov 25 2020
%p seq(coeff(series(x^4/mul(1-x^(m+4), m=0..65), x, n+1), x, n), n = 1..60); # _G. C. Greubel_, Nov 03 2019
%t Table[Count[IntegerPartitions[n],_?(Min[#]==4&)],{n,60}] (* _Harvey P. Dale_, May 13 2012 *)
%t Rest@CoefficientList[Series[x^4/QPochhammer[x^4, x], {x,0,60}], x] (* _G. C. Greubel_, Nov 03 2019 *)
%o (PARI) my(x='x+O('x^60)); concat([0,0,0], Vec(x^4/prod(m=0,70, 1-x^(m+4)))) \\ _G. C. Greubel_, Nov 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0,0,0] cat Coefficients(R!( x^4/(&*[1-x^(m+4): m in [0..70]]) )); // _G. C. Greubel_, Nov 03 2019
%o (Sage)
%o def A026797_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x^4/product((1-x^(m+4)) for m in (0..60)) ).list()
%o a=A026797_list(60); a[1:] # _G. C. Greubel_, Nov 03 2019
%Y Essentially the same as A008484.
%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), this sequence (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).
%Y Not necessarily connected k-regular simple graphs girth exactly 4: A198314 (any k), A185644 (triangle); fixed k: this sequence (k=2), A185134 (k=3), A185144 (k=4).
%K nonn,easy
%O 1,12
%A _Clark Kimberling_
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