A281357 G.f.: (Product_{j>=1} 1/(1-q^j)^2 + Product_{j>=1} 1/(1-q^(2*j)))/2.

1, 1, 3, 5, 11, 18, 34, 55, 95, 150, 244, 376, 588, 885, 1340, 1978, 2922, 4235, 6130, 8745, 12442, 17501, 24533, 34075, 47156, 64756, 88594, 120420, 163075, 219595, 294652, 393407, 523468, 693465, 915681, 1204329, 1579087, 2063035, 2687440, 3489365, 4518083, 5832448

Offset

0,3

Comments

It appears that a(n) is the number of ways you can place n unlabelled balls into any number of unlabelled bags and then place those (non-empty) bags into two unlabelled boxes. See emails to seqfan list by Rick Shepherd, Jon Wild, and Allan Wechsler on March 10 2025. - Jon Wild, Mar 11 2025

Links

Table of n, a(n) for n=0..41.

Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.

Formula

a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Oct 13 2017

a(n) = (A000712(n) + A035363(n))/2. - Alois P. Heinz, Mar 11 2025

Mathematica

CoefficientList[Series[(1/QPochhammer[x, x]^2 + 1/QPochhammer[x^2, x^2])/2, {x, 0, 50}], x] (* Vaclav Kotesovec, Oct 13 2017 *)

Prog

(PARI) nn = 50; Vec((prod(j=1, nn, 1/(1-q^j)^2) + prod(j=1, nn, 1/(1-q^(2*j))))/2 + O(q^nn)) \\ Michel Marcus, Oct 13 2017

Crossrefs

Cf. A000712, A035363.

Sequence in context: A373273 A244473 A023597 * A320789 A269628 A162891

Adjacent sequences: A281354 A281355 A281356 * A281358 A281359 A281360

Keyword

nonn,changed

Author

N. J. A. Sloane, Jan 22 2017

Extensions

More terms from Ilya Gutkovskiy, Feb 02 2017

Status

approved