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A209816

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.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

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.

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

EXAMPLE

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)

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 *)

PROG

(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

CROSSREFS