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Number of partitions of n into distinct parts such that (u+1)^2 <= v for all pairs (u,v) of parts with u<v.
5

%I #4 Mar 30 2012 18:50:58

%S 1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,

%T 6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,10,10,10,10,

%U 10,10,10,10,10,10,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12

%N Number of partitions of n into distinct parts such that (u+1)^2 <= v for all pairs (u,v) of parts with u<v.

%C A132086, A132087 and A132088 give record values and where and how often they occur.

%H R. Zumkeller, <a href="/A132085/b132085.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = f(n,1) with f(m,p) = if p=m then 1 else (if p<m then f(m-1,(p+1)^2)+f(m,p+1) else 0).

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

%e a(20)=#{20,19+1,18+2,17+3}=4;

%e a(30)=#{30,29+1,28+2,27+3,26+4,25+4+1}=6;

%e a(40)=#{40,39+1,38+2,37+3,36+4,35+4+1}=6;

%e a(50)=#{50,49+1,48+2,47+3,46+4,45+5,45+4+1,44+5+1}=8;

%e a(60)=#{60,59+1,58+2,57+3,56+4,55+5,55+4+1,54+6,54+5+1,53+6+1}=10.

%Y Cf. A000009, A132015.

%K nonn

%O 1,5

%A _Reinhard Zumkeller_, Aug 10 2007