

A132011


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


3



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
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OFFSET

1,4


COMMENTS

Also the dimension of the nth degree part of the mod 3 Steenrod algebra.
Also the number of partitions into parts (3^j1)/2=1+3+3^2+...+3^(j1) for j>=1. (End)


LINKS



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^k1)/(m1), thus g.f. for the number of such partitions is 1/Product_{k>0} (1x^((m^k1)/(m1))).  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.
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


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



