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A086737 a(n) = A000217(A000041(n)).

%I A086737 #24 Oct 27 2024 12:12:46
%S A086737 1,1,3,6,15,28,66,120,253,465,903,1596,3003,5151,9180,15576,26796,
%T A086737 44253,74305,120295,196878,314028,502503,788140,1241100,1917861,
%U A086737 2968266,4531555,6913621,10421895,15705210,23409903,34857075,51445296,75774205,110759286
%N A086737 a(n) = A000217(A000041(n)).
%C A086737 a(n) is the number of partitions of 2n that are sum-symmetric. That is, a(n) is the number of partitions of 2n that can be divided into two subsequences (no central summand) that each total to n. Example: Of the 11 partitions of 6, there are 6 that are sum-symmetric (partition subsequences bracketed [] and listed in descending order for clarity:) [3][3], [3][2,1], [3][1,1,1], [2,1][2,1], [2,1][1,1,1], [1,1,1][1,1,1]. As this example suggests, a(n) = p(n)*(p(n)+1)/2. - _Gregory L. Simay_, Oct 26 2015
%H A086737 Alois P. Heinz, <a href="/A086737/b086737.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 1..1000 from Robert Israel)
%H A086737 Tara Kalsi, Alessandro Romito, and Henning Schomerus, <a href="https://arxiv.org/abs/2410.15872">Hierarchical analytical approach to universal spectral correlations in Brownian Quantum Chaos</a>, arXiv:2410.15872 [cond-mat.mes-hall], 2024. See p. 21.
%p A086737 f:= proc(n) local p;
%p A086737   p:= combinat:-numbpart(n);
%p A086737   p*(p+1)/2
%p A086737 end proc:
%p A086737 map(f, [$1..100]); # _Robert Israel_, Oct 26 2015
%t A086737 pp = Array[PartitionsP, 40, 0]; pp (pp + 1)/2 (* _Jean-François Alcover_, Mar 19 2019 *)
%o A086737 (PARI) a(n) = apply(x->x*(x+1)/2, numbpart(n)); \\ _Michel Marcus_, Oct 26 2015
%Y A086737 Cf. A000041, A000217, A108796.
%K A086737 nonn
%O A086737 0,3
%A A086737 _Jon Perry_, Jul 29 2003
%E A086737 a(0)=1 prepended by _Alois P. Heinz_, Mar 25 2017