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
FORMULA
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Mar 13 2012
EXTENSIONS
More terms from Alois P. Heinz, Jul 09 2012
STATUS
approved