OFFSET
0,3
COMMENTS
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
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..1000 from Robert Israel)
Tara Kalsi, Alessandro Romito, and Henning Schomerus, Hierarchical analytical approach to universal spectral correlations in Brownian Quantum Chaos, arXiv:2410.15872 [cond-mat.mes-hall], 2024. See p. 21.
MAPLE
f:= proc(n) local p;
p:= combinat:-numbpart(n);
p*(p+1)/2
end proc:
map(f, [$1..100]); # Robert Israel, Oct 26 2015
MATHEMATICA
pp = Array[PartitionsP, 40, 0]; pp (pp + 1)/2 (* Jean-François Alcover, Mar 19 2019 *)
PROG
(PARI) a(n) = apply(x->x*(x+1)/2, numbpart(n)); \\ Michel Marcus, Oct 26 2015
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Jon Perry, Jul 29 2003
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Mar 25 2017
STATUS
approved