A349675 a(n) is the number of attainable partitions of n.

1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 9, 9, 13, 13, 17, 17, 21, 21, 27, 27, 34, 34, 41, 41, 51, 51, 62, 62, 73, 73, 88, 88, 105, 105, 122, 122, 144, 144, 168, 168, 193, 193, 225, 225, 260, 260, 296, 296, 340, 340, 388, 388, 438, 438, 498, 498, 564, 564, 632, 632, 713, 713

Offset

0,3

Comments

An attainable partition p of n is a partition such that, when written so that p_1 >= p_2 >= ... >= p_r, we have Sum_{i=1..r} (3-2*i)*p_i >= 0.

Links

Table of n, a(n) for n=0..61.

Kathleen Petersen and James Sellers, Partitions Associated to Class Groups of Imaginary Quadratic Number Fields, arXiv:2111.12031 [math.NT], 2021.

Formula

G.f.: (1/(1-x))*Product_{i>=1} 1/(1-x^(i*(i+1)). See Theorem 1.1 p. 1.

Mathematica

nterms=50; Table[Total[Map[If[Sum[(3-2i)#[[i]], {i, Length[#]}]>=0, 1, 0]&, IntegerPartitions[n]]], {n, 0, nterms-1}] (* Paolo Xausa, Nov 24 2021 *)

Prog

(PARI) a(n) = my(nb=0); forpart(p=n, p = vecsort(p, , 4); if (sum(i=1, #p, (3-2*i)*p[i]) >= 0, nb++); ); nb;

Crossrefs

Cf. A000041, A302835 (bisection).

Sequence in context: A309683 A283529 A064986 * A029019 A040039 A008667

Adjacent sequences: A349672 A349673 A349674 * A349676 A349677 A349678

Keyword

nonn

Author

Michel Marcus, Nov 24 2021

Status

approved