A269628 Dimension of BSym_n.

1, 1, 3, 5, 11, 18, 35, 57, 102, 165, 279, 444, 726, 1136, 1804, 2785, 4326, 6584, 10048, 15100, 22698, 33723, 50034, 73557, 107912, 157122, 228189, 329341, 473998, 678576, 968672, 1376402, 1950177, 2751900, 3872346, 5429166, 7591294, 10579486, 14705595, 20379419, 28172006, 38836332, 53410265, 73264431, 100271052

Offset: 0

Author

Eric S. Egge, Mar 01 2016

Keywords

• nonn

Comments

BSym_n is the space of homogeneous series of degree n in the variables x_1, x_{-1}, x_2, x_{-2}, ... which are invariant under the natural action of the hyperoctahedral group.

a(n) is also the number of Ferrers diagrams (in the English convention) in which some boxes contain a dot, such that the dots are left-justified in each row, and for each k, the dots in rows with k boxes form a Ferrers shape, and there are n total dots and boxes.

a(n) is also the number of partitions of n in which there are 1 + floor(k/2) different parts of "type" k for each k.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Vaclav Kotesovec)

Crossrefs

Cf. A275416.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(add(d*ceil(
      (d+1)/2), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Sep 20 2017

Mathematica

Table[SeriesCoefficient[1/Product[(1 - x^j)^Floor[(j + 2)/2], {j, 1, n}], {x, 0, n}], {n, 0, 44}] (* Michael De Vlieger, Mar 06 2016 *)

PARI

N=66;  x='x+O('x^N); Vec( 1 / prod(j=1, N, (1-x^j)^floor((j+2)/2) ) ) \\ Joerg Arndt, Mar 02 2016

Formula

G.f.: 1 / (Product_{j>=1} (1-x^j)^floor((j+2)/2)).

a(n) ~ Zeta(3)^(25/72) / (A^(1/2) * Pi * 2^(3/4) * sqrt(3) * n^(61/72)) * exp(1/24 - Pi^4/(384*Zeta(3)) + Pi^2*n^(1/3) / (8*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 06 2016

a(n) = A275416(2n,n). - Alois P. Heinz, Sep 19 2017