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A117989
Number of partitions of n such that the least part occurs at least twice.
24
0, 1, 1, 3, 3, 7, 8, 14, 18, 28, 35, 53, 67, 94, 121, 165, 209, 280, 353, 462, 582, 749, 935, 1192, 1480, 1862, 2302, 2871, 3526, 4366, 5335, 6555, 7976, 9737, 11789, 14317, 17259, 20845, 25032, 30093, 35992, 43087, 51347, 61216, 72710, 86362, 102235
OFFSET
1,4
COMMENTS
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].
Also, the number of partitions of 2n such that the difference between greatest part and smallest part is n. - Vladeta Jovovic, May 09 2008
LINKS
Aritram Dhar, Proofs of Two Formulas of Vladeta Jovovic, arXiv:2112.07762 [math.CO], 2021.
FORMULA
G.f.: sum(k>=1, x^(2*k)/prod(j>=k, 1-x^j ) ).
G.f.: sum(k>=1, x^k*(1-x^(k-1))/prod(j=1..k, 1-x^j ) ).
a(n) = 2*A000041(n) - A000041(n+1). - Vladeta Jovovic, Jul 21 2006
a(n) = A056823(n+1) - 2*A056823(n). - Bob Selcoe, Apr 11 2014
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
EXAMPLE
a(5) = 3 because we have [3,1,1], [2,1,1,1] and [1,1,1,1,1].
MAPLE
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);
A117989 := proc(n)
2*combinat[numbpart](n)-combinat[numbpart](n+1) ;
end proc: # R. J. Mathar, May 19 2016
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Count[#, Min[#]]>1&]], {n, 50}] (* Harvey P. Dale, Apr 23 2011 *)
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 *)
PROG
(Haskell)
a117989 n = a117989_list !! (n-1)
a117989_list = tail $ zipWith (-)
(map (* 2) a000041_list) $ tail a000041_list
-- Reinhard Zumkeller, Nov 12 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 08 2006
STATUS
approved