|
%I #73 Sep 08 2022 08:44:49
%S 0,0,0,1,0,0,1,1,1,2,2,3,4,5,6,9,10,13,17,21,25,33,39,49,60,73,88,110,
%T 130,158,191,230,273,331,391,468,556,660,779,927,1087,1284,1510,1775,
%U 2075,2438,2842,3323,3872,4510,5237,6095,7056,8182,9465,10945,12625
%N Number of partitions of n in which the least part is 3.
%C Let b(k) be the number of partitions of k for which twice the number of ones is the number of parts, k = 0, 1, 2, ... . Then a(n+4) = b(n), n = 0, 1, 2, ... (conjectured). - _George Beck_, Aug 19 2017
%H Vaclav Kotesovec, <a href="/A026796/b026796.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%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^3 / Product_{m>=3} (1 - x^m).
%F a(n) = p(n-3) - p(n-4) - p(n-5) + p(n-6), where p(n) = A000041(n). - _Bob Selcoe_, Aug 07 2014
%F a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^2 / (12*sqrt(3)*n^2). - _Vaclav Kotesovec_, Jun 02 2018
%F G.f.: Sum_{k>=1} x^(3*k) / Product_{j=1..k-1} (1 - x^j). - _Ilya Gutkovskiy_, Nov 25 2020
%p seq(coeff(series(x^3/mul(1-x^(m+3), m=0..65), x, n+1), x, n), n = 0 .. 60); # _G. C. Greubel_, Nov 02 2019
%t Table[Count[IntegerPartitions[n], p_ /; Min@p==3], {n, 0, 60}] (* _George Beck_ Aug 19 2017 *)
%t CoefficientList[Series[x^3/QPochhammer[x^3, x], {x,0,60}], x] (* _G. C. Greubel_, Nov 02 2019 *)
%o (PARI) a(n) = numbpart(n-3) - numbpart(n-4) - numbpart(n-5) + numbpart(n-6); \\ _Michel Marcus_, Aug 20 2014
%o (PARI) x='x+O('x^66); Vecrev(Pol(x^3*(1-x)*(1-x^2)/eta(x))) \\ _Joerg Arndt_, Aug 22 2014
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0,0,0] cat Coefficients(R!( x^3/(&*[1-x^(m+3): m in [0..70]]) )); // _G. C. Greubel_, Nov 02 2019
%o (Sage)
%o def A026796_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x^3/product((1-x^(m+3)) for m in (0..65)) ).list()
%o A026796_list(60) # _G. C. Greubel_, Nov 02 2019
%Y Essentially the same sequence as A008483.
%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), this sequence (g=3), A026797 (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 3: A198313 (any k), A185643 (triangle); fixed k: this sequence (k=2), A185133 (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).
%K nonn,easy
%O 0,10
%A _Clark Kimberling_
%E More terms from _Michel Marcus_, Aug 20 2014
%E a(0) = 0 prepended by _Joerg Arndt_, Aug 22 2014
|
