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

1, 1, 3, 6, 15, 28, 66, 120, 253, 465, 903, 1596, 3003, 5151, 9180, 15576, 26796, 44253, 74305, 120295, 196878, 314028, 502503, 788140, 1241100, 1917861, 2968266, 4531555, 6913621, 10421895, 15705210, 23409903, 34857075, 51445296, 75774205, 110759286

Offset: 0

Author

Jon Perry, Jul 29 2003

Keywords

• nonn

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.

Crossrefs

Cf. A000041, A000217, A108796.

Programs

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 *)

PARI

a(n) = apply(x->x*(x+1)/2, numbpart(n)); \\ Michel Marcus, Oct 26 2015

Extensions

a(0)=1 prepended by Alois P. Heinz, Mar 25 2017