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

%I A132011 #18 Jun 24 2024 22:32:55
%S A132011 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 A132011 15,16,17,18,18,19,20,21,22,24,25,26,27,29,30,31,32,34,35,36,37,40,42,
%U A132011 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 A132011 Number of partitions of n into distinct parts such that 3*u<=v for all pairs (u,v) of parts with u<v.
%C A132011 From _Edward Early_, Jan 10 2009: (Start)
%C A132011 Also the dimension of the n-th degree part of the mod 3 Steenrod algebra.
%C A132011 Also the number of partitions into parts (3^j-1)/2=1+3+3^2+...+3^(j-1) for j>=1. (End)
%H A132011 Reinhard Zumkeller, <a href="/A132011/b132011.txt">Table of n, a(n) for n = 1..1000</a>
%F A132011 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 A132011 a(10) = #{10, 9+1, 8+2} = 3;
%e A132011 a(11) = #{11, 10+1, 9+2} = 3;
%e A132011 a(12) = #{12, 11+1, 10+2, 9+3} = 4;
%e A132011 a(13) = #{13, 12+1, 11+2, 10+3, 9+3+1} = 5.
%e A132011 From _Joerg Arndt_, Dec 28 2012: (Start)
%e A132011 The a(33)=17 such partitions of 33 are
%e A132011 [ 1]  [ 24 7 2 ]
%e A132011 [ 2]  [ 24 8 1 ]
%e A132011 [ 3]  [ 25 6 2 ]
%e A132011 [ 4]  [ 25 7 1 ]
%e A132011 [ 5]  [ 25 8 ]
%e A132011 [ 6]  [ 26 6 1 ]
%e A132011 [ 7]  [ 26 7 ]
%e A132011 [ 8]  [ 27 5 1 ]
%e A132011 [ 9]  [ 27 6 ]
%e A132011 [10]  [ 28 4 1 ]
%e A132011 [11]  [ 28 5 ]
%e A132011 [12]  [ 29 3 1 ]
%e A132011 [13]  [ 29 4 ]
%e A132011 [14]  [ 30 3 ]
%e A132011 [15]  [ 31 2 ]
%e A132011 [16]  [ 32 1 ]
%e A132011 [17]  [ 33 ]
%e A132011 (End)
%o A132011 (Haskell)
%o A132011 a132011 = p [1..] where
%o A132011    p _  0 = 1
%o A132011    p (k:ks) m = if m < k then 0 else p [3 * k ..] (m - k) + p ks m
%o A132011 -- _Reinhard Zumkeller_, Oct 10 2013
%Y A132011 Cf. A000009, A000929.
%Y A132011 Cf. A147583. - _Reinhard Zumkeller_, Nov 08 2008
%K A132011 nonn
%O A132011 1,4
%A A132011 _Reinhard Zumkeller_, Aug 07 2007