A132011 Number of partitions of n into distinct parts such that 3*u<=v for all pairs (u,v) of parts with u<v.

1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7, 8, 9, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 40, 42, 43, 44, 47, 49, 50, 51, 54, 56, 57, 58, 61, 64, 66, 67, 70, 73, 75, 76, 79, 82, 84, 85, 88, 91

Offset

1,4

Comments

From Edward Early, Jan 10 2009: (Start)

Also the dimension of the n-th degree part of the mod 3 Steenrod algebra.

Also the number of partitions into parts (3^j-1)/2=1+3+3^2+...+3^(j-1) for j>=1. (End)

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Formula

More generally, number of partitions of n into distinct parts such that m*u<=v for all pairs (u,v) of parts with u<v is equal to the number of partitions of n into parts of the form (m^k-1)/(m-1), thus g.f. for the number of such partitions is 1/Product_{k>0} (1-x^((m^k-1)/(m-1))). - Vladeta Jovovic, Jan 09 2009

Example

a(10) = #{10, 9+1, 8+2} = 3;
a(11) = #{11, 10+1, 9+2} = 3;
a(12) = #{12, 11+1, 10+2, 9+3} = 4;
a(13) = #{13, 12+1, 11+2, 10+3, 9+3+1} = 5.
From Joerg Arndt, Dec 28 2012: (Start)
The a(33)=17 such partitions of 33 are
[ 1]  [ 24 7 2 ]
[ 2]  [ 24 8 1 ]
[ 3]  [ 25 6 2 ]
[ 4]  [ 25 7 1 ]
[ 5]  [ 25 8 ]
[ 6]  [ 26 6 1 ]
[ 7]  [ 26 7 ]
[ 8]  [ 27 5 1 ]
[ 9]  [ 27 6 ]
[10]  [ 28 4 1 ]
[11]  [ 28 5 ]
[12]  [ 29 3 1 ]
[13]  [ 29 4 ]
[14]  [ 30 3 ]
[15]  [ 31 2 ]
[16]  [ 32 1 ]
[17]  [ 33 ]
(End)

Prog

 (Haskell)
a132011 = p [1..] where
   p _  0 = 1
   p (k:ks) m = if m < k then 0 else p [3 * k ..] (m - k) + p ks m
-- Reinhard Zumkeller, Oct 10 2013

Crossrefs

Cf. A000009, A000929.

Cf. A147583. - Reinhard Zumkeller, Nov 08 2008

Sequence in context: A088004 A070548 A209628 * A363822 A054893 A090617

Adjacent sequences: A132008 A132009 A132010 * A132012 A132013 A132014

Keyword

nonn

Author

Reinhard Zumkeller, Aug 07 2007

Status

approved