A209816 Number of partitions of 2n in which every part is

1, 3, 7, 15, 30, 58, 105, 186, 318, 530, 863, 1380, 2164, 3345, 5096, 7665, 11395, 16765, 24418, 35251, 50460, 71669, 101050, 141510, 196888, 272293, 374423, 512081, 696760, 943442, 1271527, 1706159, 2279700, 3033772, 4021695, 5311627, 6990367, 9168321

Offset: 1

Author

Clark Kimberling, Mar 13 2012

Keywords

• nonn

Comments

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

Also, the number of partitions of 3n into n parts. - Seiichi Manyama, May 07 2018

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

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

Examples

			The 7 partitions of 6 with parts <4 are as follows:
3+3, 3+2+1, 3+1+1+1
2+2+2, 2+2+1+1, 2+1+1+1+1
1+1+1+1+1+1.
Matching partitions of 2 into rationals as described:
1 + 1
1 + 3/3 + 1/3
1 + 1/3 + 1/3 + 1/3
2/3 + 2/3 + 2/3
2/3 + 2/3 + 1/3 + 1/3
2/3 + 1/3 + 1/3 + 1/3 + 1/3
1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3.
From _Seiichi Manyama_, May 07 2018: (Start)
n | Partitions of 3n into n parts
--+-------------------------------------------------
1 | 3;
2 | 5+1, 4+2, 3+3;
3 | 7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3; (End)
From _Gus Wiseman_, Oct 24 2018: (Start)
The a(1) = 1 through a(4) = 15 partitions:
  (11)  (22)    (33)      (44)
        (211)   (222)     (332)
        (1111)  (321)     (422)
                (2211)    (431)
                (3111)    (2222)
                (21111)   (3221)
                (111111)  (3311)
                          (4211)
                          (22211)
                          (32111)
                          (41111)
                          (221111)
                          (311111)
                          (2111111)
                          (11111111)
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Gus Wiseman, Multigraphs realizing each of the a(4) = 15 multigraphical partitions of 8.
• Gus Wiseman, Multigraphs realizing each of the a(5) = 30 multigraphical partitions of 10.

Crossrefs

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

Programs

Haskell

a209816 n = p [1..n] (2*n) where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 14 2013

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 09 2012

Mathematica

f[n_] := Length[Select[IntegerPartitions[2 n], First[#] <= n &]]; Table[f[n], {n, 1, 30}] (* A209816 *)
Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,2*n}],{n,1,20}] (* Vaclav Kotesovec, May 25 2015 *)
Table[Length@IntegerPartitions[3n, {n}], {n, 25}] (* Vladimir Reshetnikov, Jul 24 2016 *)
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 *)

Formula

a(n) = A000041(2*n)-A000070(n-1). - Matthew Vandermast, Jul 16 2012

a(n) = Sum_{k=1..n} A008284(2*n, k) = A000041(2*n) - A000070(n-1), for n >= 1. - Wolfdieter Lang, May 21 2019

Extensions

More terms from Alois P. Heinz, Jul 09 2012