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Number of partitions of 2n in which every part is <n+1; also, the number of partitions of 2 into rational numbers a/b such that 0<a<=b<=n and b divides n.
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%I #62 Oct 30 2021 15:04:46

%S 1,3,7,15,30,58,105,186,318,530,863,1380,2164,3345,5096,7665,11395,

%T 16765,24418,35251,50460,71669,101050,141510,196888,272293,374423,

%U 512081,696760,943442,1271527,1706159,2279700,3033772,4021695,5311627,6990367,9168321

%N Number of partitions of 2n in which every part is <n+1; also, the number of partitions of 2 into rational numbers a/b such that 0<a<=b<=n and b divides n.

%C Also, the number of partitions of 3n in which n is the maximal part.

%C Also, the number of partitions of 3n into n parts. - _Seiichi Manyama_, May 07 2018

%C Also the number of multigraphical partitions of 2n, i.e., integer partitions that comprise the multiset of vertex-degrees of some multigraph. - _Gus Wiseman_, Oct 24 2018

%C Also number of partitions of 2n with at most n parts. Conjugate partitions map one to one to partitions of 2*n with each part <= n. - _Wolfdieter Lang_, May 21 2019

%H Alois P. Heinz, <a href="/A209816/b209816.txt">Table of n, a(n) for n = 1..1000</a>

%H Gus Wiseman, <a href="/A209816/a209816.png">Multigraphs realizing each of the a(4) = 15 multigraphical partitions of 8.</a>

%H Gus Wiseman, <a href="/A209816/a209816_1.png">Multigraphs realizing each of the a(5) = 30 multigraphical partitions of 10.</a>

%F a(n) = A000041(2*n)-A000070(n-1). - _Matthew Vandermast_, Jul 16 2012

%F a(n) = Sum_{k=1..n} A008284(2*n, k) = A000041(2*n) - A000070(n-1), for n >= 1. - _Wolfdieter Lang_, May 21 2019

%e The 7 partitions of 6 with parts <4 are as follows:

%e 3+3, 3+2+1, 3+1+1+1

%e 2+2+2, 2+2+1+1, 2+1+1+1+1

%e 1+1+1+1+1+1.

%e Matching partitions of 2 into rationals as described:

%e 1 + 1

%e 1 + 3/3 + 1/3

%e 1 + 1/3 + 1/3 + 1/3

%e 2/3 + 2/3 + 2/3

%e 2/3 + 2/3 + 1/3 + 1/3

%e 2/3 + 1/3 + 1/3 + 1/3 + 1/3

%e 1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3.

%e From _Seiichi Manyama_, May 07 2018: (Start)

%e n | Partitions of 3n into n parts

%e --+-------------------------------------------------

%e 1 | 3;

%e 2 | 5+1, 4+2, 3+3;

%e 3 | 7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3; (End)

%e From _Gus Wiseman_, Oct 24 2018: (Start)

%e The a(1) = 1 through a(4) = 15 partitions:

%e (11) (22) (33) (44)

%e (211) (222) (332)

%e (1111) (321) (422)

%e (2211) (431)

%e (3111) (2222)

%e (21111) (3221)

%e (111111) (3311)

%e (4211)

%e (22211)

%e (32111)

%e (41111)

%e (221111)

%e (311111)

%e (2111111)

%e (11111111)

%e (End)

%p b:= proc(n, i) option remember;

%p `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))

%p end:

%p a:= n-> b(2*n, n):

%p seq(a(n), n=1..50); # _Alois P. Heinz_, Jul 09 2012

%t f[n_] := Length[Select[IntegerPartitions[2 n], First[#] <= n &]]; Table[f[n], {n, 1, 30}] (* A209816 *)

%t Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,2*n}],{n,1,20}] (* _Vaclav Kotesovec_, May 25 2015 *)

%t Table[Length@IntegerPartitions[3n, {n}], {n, 25}] (* _Vladimir Reshetnikov_, Jul 24 2016 *)

%t b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[2*n, n]; Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)

%o (Haskell)

%o a209816 n = p [1..n] (2*n) where

%o p _ 0 = 1

%o p [] _ = 0

%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

%o -- _Reinhard Zumkeller_, Nov 14 2013

%Y Cf. A000041, A000070, A000569, A008284, A025065, A079122, A096373, A147878, A209815, A320911, A320921, A320924.

%K nonn

%O 1,2

%A _Clark Kimberling_, Mar 13 2012

%E More terms from _Alois P. Heinz_, Jul 09 2012