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A179822 Maximally refined partitions into distinct parts (of any natural number) with largest part n.

%I A179822 #11 Apr 13 2021 19:21:19
%S A179822 1,1,2,3,5,7,12,16,26,37,58,79,128,171,271,376,576,783,1239,1654,2567,
%T A179822 3505,5382,7245,11247,15036,23187,31370,47672,64146,98887,131784,
%U A179822 201340,271350,412828,551744,843285,1125417,1715207,2299452,3479341,4654468,7090529
%N A179822 Maximally refined partitions into distinct parts (of any natural number) with largest part n.
%C A179822 For the definition, see sequence A179009. This sequence counts the same objects using a different statistic, the largest part rather than the sum of the parts.
%C A179822 a(n) is the number of subsets of {1..n-1} containing the sum of any two distinct elements whose sum is <= n. This differs from A326080 in that the set may not contain n itself. These sets are the complements of the set of parts in the first definition. - _Andrew Howroyd_, Apr 13 2021
%e A179822 The partitions counted by n=4 are:
%e A179822   4+1, 4+2+1, 4+3+1, 4+3+2, 4+3+2+1.
%e A179822 The partitions counted by n=5 are:
%e A179822   5+2+1, 5+3+1, 5+3+2+1, 5+4+2+1, 5+4+3+1, 5+4+3+2, 5+4+3+2+1.
%o A179822 (PARI)
%o A179822 a(n)={
%o A179822   my(ok(k,b)=for(i=1, (k-1)\2, if(bittest(b,i) && bittest(b,k-i), return(0))); 1);
%o A179822   my(recurse(k,b)=if(k==n, ok(k,b), self()(k+1, bitor(b,1<<k)) + if(ok(k,b), self()(k+1, b))));
%o A179822   if(n<1, n==0, recurse(1, 0))
%o A179822 } \\ _Andrew Howroyd_, Apr 13 2021
%Y A179822 Cf. A179009, A179817, A326080.
%K A179822 nonn
%O A179822 0,3
%A A179822 _Moshe Shmuel Newman_, Jan 10 2011
%E A179822 a(19)-a(42) from _Andrew Howroyd_, Apr 13 2021