A269628 - OEIS

%I #20 Sep 20 2017 10:04:31

%S 1,1,3,5,11,18,35,57,102,165,279,444,726,1136,1804,2785,4326,6584,

%T 10048,15100,22698,33723,50034,73557,107912,157122,228189,329341,

%U 473998,678576,968672,1376402,1950177,2751900,3872346,5429166,7591294,10579486,14705595,20379419,28172006,38836332,53410265,73264431,100271052

%N Dimension of BSym_n.

%C 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.

%C 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.

%C 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.

%H Alois P. Heinz, <a href="/A269628/b269628.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from Vaclav Kotesovec)

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

%F 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

%F a(n) = A275416(2n,n). - _Alois P. Heinz_, Sep 19 2017

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*ceil(

%p       (d+1)/2), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..45);  # _Alois P. Heinz_, Sep 20 2017

%t 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 *)

%o (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

%Y Cf. A275416.

%K nonn

%O 0,3

%A _Eric S. Egge_, Mar 01 2016