|
%I #40 Sep 08 2022 08:44:49
%S 1,0,0,0,0,1,0,0,0,0,1,1,1,1,1,2,2,3,3,4,5,6,7,9,10,13,15,18,21,26,30,
%T 36,42,50,58,70,80,95,110,129,150,176,202,236,272,317,364,423,484,560,
%U 643,740,847,975,1112,1277,1456,1666,1897,2168
%N Number of partitions of n in which the least part is 5.
%C Also the number of not necessarily connected 2-regular simple graphs with girth exactly 5. - _Jason Kimberley_, Nov 11 2011
%C Such partitions of n+5 correspond to A185325 partitions (parts >= 5) of n by removing a single part of size 5. - _Jason Kimberley_, Nov 11 2011
%H G. C. Greubel, <a href="/A026798/b026798.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^5 * Product_{m>=5} 1/(1-x^m).
%F a(n+5) is given by p(n) - p(n-1) - p(n-2) + 2p(n-5) - p(n-8) - p(n-9) + p(n-10) where p(n) = A000041(n). - _Shanzhen Gao_, Oct 28 2010 [sign of 10 and offset of formula corrected by _Jason Kimberley_, Nov 11 2011]
%F a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^4 / (6*sqrt(3)*n^3). - _Vaclav Kotesovec_, Jun 02 2018
%p ZL := [ B,{B=Set(Set(Z, card>=5))}, unlabeled ]: 1,0,0,0,0, seq(combstruct[count](ZL, size=n), n=0..54); # _Zerinvary Lajos_, Mar 13 2007
%p 1, seq(coeff(series(x^5/mul(1-x^(m+5), m=0..70), x, n+1), x, n), n = 0..65); # _G. C. Greubel_, Nov 03 2019
%t f[1, 1] = 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[{1, 0, 0, 0, 0, 1}, Table[ f[n, 5], {n, 50}]] (* _Robert G. Wilson v_ *)
%t Join[{1}, Drop[CoefficientList[Series[x^5/QPochhammer[x^5, x], {x,0,60}], x], 1]] (* _G. C. Greubel_, Nov 03 2019 *)
%o (PARI) my(x='x+O('x^60)); concat([1,0,0,0,0], Vec(x^5/prod(m=0,70, 1-x^(m+5)))) \\ _G. C. Greubel_, Nov 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [1,0,0,0,0] cat Coefficients(R!( x^5/(&*[1-x^(m+5): m in [0..70]]) )); // _G. C. Greubel_, Nov 03 2019
%o (Sage)
%o def A026798_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x^5/product((1-x^(m+5)) for m in (0..70)) ).list()
%o a=A026798_list(65); [1]+a[1:] # _G. C. Greubel_, Nov 03 2019
%Y Essentially the same as A185325.
%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), this sequence (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - _Jason Kimberley_, Nov 11 2011
%K nonn,easy
%O 0,16
%A _Clark Kimberling_
|
