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Number of partitions of n into distinct parts such that 3*u<=v for all pairs (u,v) of parts with u<v.
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%I #18 Jun 24 2024 22:32:55

%S 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,

%T 15,16,17,18,18,19,20,21,22,24,25,26,27,29,30,31,32,34,35,36,37,40,42,

%U 43,44,47,49,50,51,54,56,57,58,61,64,66,67,70,73,75,76,79,82,84,85,88,91

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

%C From _Edward Early_, Jan 10 2009: (Start)

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

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

%H Reinhard Zumkeller, <a href="/A132011/b132011.txt">Table of n, a(n) for n = 1..1000</a>

%F 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

%e a(10) = #{10, 9+1, 8+2} = 3;

%e a(11) = #{11, 10+1, 9+2} = 3;

%e a(12) = #{12, 11+1, 10+2, 9+3} = 4;

%e a(13) = #{13, 12+1, 11+2, 10+3, 9+3+1} = 5.

%e From _Joerg Arndt_, Dec 28 2012: (Start)

%e The a(33)=17 such partitions of 33 are

%e [ 1] [ 24 7 2 ]

%e [ 2] [ 24 8 1 ]

%e [ 3] [ 25 6 2 ]

%e [ 4] [ 25 7 1 ]

%e [ 5] [ 25 8 ]

%e [ 6] [ 26 6 1 ]

%e [ 7] [ 26 7 ]

%e [ 8] [ 27 5 1 ]

%e [ 9] [ 27 6 ]

%e [10] [ 28 4 1 ]

%e [11] [ 28 5 ]

%e [12] [ 29 3 1 ]

%e [13] [ 29 4 ]

%e [14] [ 30 3 ]

%e [15] [ 31 2 ]

%e [16] [ 32 1 ]

%e [17] [ 33 ]

%e (End)

%o (Haskell)

%o a132011 = p [1..] where

%o p _ 0 = 1

%o p (k:ks) m = if m < k then 0 else p [3 * k ..] (m - k) + p ks m

%o -- _Reinhard Zumkeller_, Oct 10 2013

%Y Cf. A000009, A000929.

%Y Cf. A147583. - _Reinhard Zumkeller_, Nov 08 2008

%K nonn

%O 1,4

%A _Reinhard Zumkeller_, Aug 07 2007