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%I #48 Dec 27 2023 19:24:20
%S 0,0,0,0,0,0,1,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,4,4,6,6,8,9,11,12,16,17,
%T 21,24,29,32,40,44,53,60,71,80,96,107,126,143,167,188,221,248,288,326,
%U 376,424,491,552,634,716,819,922,1056,1187,1353,1523,1730,1944,2209,2478,2806,3151
%N Number of partitions of n in which the least part is 6.
%C a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth exactly 6 (all such graphs are simple). Each integer part i corresponds to an i-cycle; the addition of integers corresponds to the disconnected union of cycles.
%H G. C. Greubel, <a href="/A026799/b026799.txt">Table of n, a(n) for n = 0..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^6 * Product_{m>=6} 1/(1-x^m).
%F a(n) = p(n-6) -p(n-7) -p(n-8) +p(n-11) +p(n-12) +p(n-13) -p(n-14) -p(n-15) -p(n-16) +p(n-19) +p(n-20) -p(n-21) for n>0 where p(n) = A000041(n). - _Shanzhen Gao_, Oct 28 2010
%F a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^5 / (18*sqrt(2)*n^(7/2)). - _Vaclav Kotesovec_, Jun 02 2018
%F G.f.: Sum_{k>=1} x^(6*k) / Product_{j=1..k-1} (1 - x^j). - _Ilya Gutkovskiy_, Nov 25 2020
%e a(0)=0 because there does not exist a least part of the empty partition.
%e The a(6)=1 partition is 6.
%e The a(12)=1 partition is 6+6.
%e The a(13)=1 partition is 6+7.
%e .............................
%e The a(17)=1 partition is 6+11.
%e The a(18)=2 partitions are 6+6+6 and 6+12.
%p ZL := [ B,{B=Set(Set(Z, card>=6))}, unlabeled ]: 0,0,0,0,0,0, seq(combstruct[count](ZL, size=n), n=0..63); # _Zerinvary Lajos_, Mar 13 2007
%p seq(coeff(series(x^6/mul(1-x^(m+6), m=0..70), x, n+1), x, n), n = 0..65); # _G. C. Greubel_, Nov 03 2019
%t f[1, 1]=f[0, k_]=1; f[n_, k_]:= f[n, k] = If[n<0, 0, If[k>n, 0, If[k==n, 1, f[n, k+1] +f[n-k, k]]]]; Join[{0,0,0,0,0,0}, Table[f[n, 6], {n, 0, 65}]] (* _Robert G. Wilson v_, Jan 31 2011 *)
%t CoefficientList[Series[x^6/QPochhammer[x^6, x], {x,0,70}], x] (* _G. C. Greubel_, Nov 03 2019 *)
%t Join[{0},Table[Count[IntegerPartitions[n][[;;,-1]],6],{n,70}]] (* _Harvey P. Dale_, Dec 27 2023 *)
%o (Magma) p := func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
%o A026799 := func< n | p(n-6)-p(n-7)-p(n-8)+p(n-11)+p(n-12)+p(n-13)- p(n-14)-p(n-15)-p(n-16)+p(n-19)+p(n-20)-p(n-21) >; // _Jason Kimberley_, Feb 04 2011
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0,0,0,0,0,0] cat Coefficients(R!( x^6/(&*[1-x^(m+6): m in [0..70]]) )); // _G. C. Greubel_, Nov 03 2019
%o (PARI) my(x='x+O('x^60)); concat([0,0,0,0,0,0], Vec(x^6/prod(m=0,70, 1-x^(m+6)))) \\ _G. C. Greubel_, Nov 03 2019
%o (Sage)
%o def A026799_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x^6/product((1-x^(m+6)) for m in (0..70)) ).list()
%o A026799_list(65) # _G. C. Greubel_, Nov 03 2019
%Y Essentially the same as A185326.
%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), this sequence (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - _Jason Kimberley_, Feb 04 2011
%K nonn,easy
%O 0,19
%A _Clark Kimberling_
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