|
%I #44 Dec 16 2021 04:11:30
%S 0,1,1,3,3,7,8,14,18,28,35,53,67,94,121,165,209,280,353,462,582,749,
%T 935,1192,1480,1862,2302,2871,3526,4366,5335,6555,7976,9737,11789,
%U 14317,17259,20845,25032,30093,35992,43087,51347,61216,72710,86362,102235
%N Number of partitions of n such that the least part occurs at least twice.
%C More generally, the g.f. for the number of partitions of n such that the least part occurs at least m times is sum(x^(mk)/product(1-x^j, j=k..infinity), k=1..infinity). Also, the number of partitions of n such that if k is the largest part, then k>=2 and k-1 does not occur. Example: a(5)=3 because we have [5],[4,1] and [3,1,1].
%C Also, the number of partitions of 2n such that the difference between greatest part and smallest part is n. - _Vladeta Jovovic_, May 09 2008
%H Alois P. Heinz, <a href="/A117989/b117989.txt">Table of n, a(n) for n = 1..1000</a>
%H Aritram Dhar, <a href="https://arxiv.org/abs/2112.07762">Proofs of Two Formulas of Vladeta Jovovic</a>, arXiv:2112.07762 [math.CO], 2021.
%F G.f.: sum(k>=1, x^(2*k)/prod(j>=k, 1-x^j ) ).
%F G.f.: sum(k>=1, x^k*(1-x^(k-1))/prod(j=1..k, 1-x^j ) ).
%F a(n) = 2*A000041(n) - A000041(n+1). - _Vladeta Jovovic_, Jul 21 2006
%F a(n) = A056823(n+1) - 2*A056823(n). - _Bob Selcoe_, Apr 11 2014
%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 25*Pi/(24*sqrt(6))) / sqrt(n)). - _Vaclav Kotesovec_, Nov 03 2020
%e a(5) = 3 because we have [3,1,1], [2,1,1,1] and [1,1,1,1,1].
%p g:=sum(x^k*(1-x^(k-1))/product(1-x^j,j=1..k),k=2..70): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..50);
%p A117989 := proc(n)
%p 2*combinat[numbpart](n)-combinat[numbpart](n+1) ;
%p end proc: # _R. J. Mathar_, May 19 2016
%t Table[Length[Select[IntegerPartitions[n],Count[#,Min[#]]>1&]], {n,50}] (* _Harvey P. Dale_, Apr 23 2011 *)
%t max = 48; Sum[x^(2*k)/Product[1 - x^j, {j, k, Infinity}], {k, 1, Ceiling[ max/2]}] + O[x]^max // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Sep 11 2017 *)
%o (Haskell)
%o a117989 n = a117989_list !! (n-1)
%o a117989_list = tail $ zipWith (-)
%o (map (* 2) a000041_list) $ tail a000041_list
%o -- _Reinhard Zumkeller_, Nov 12 2015
%Y Cf. A096373, A000041, A056823.
%Y Cf. A097364, A323111.
%K nonn
%O 1,4
%A _Emeric Deutsch_, Apr 08 2006
|
