A084239 Rank of K-groups of Furstenberg transformation group C*-algebras of n-torus.

1, 2, 3, 4, 6, 8, 13, 20, 32, 52, 90, 152, 268, 472, 845, 1520, 2766, 5044, 9277, 17112, 31724, 59008, 110162, 206260, 387282, 729096, 1375654, 2601640, 4929378, 9358944, 17797100, 33904324, 64678112, 123580884, 236413054, 452902072

Offset

0,2

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..500

K. Reihani, C*-algebras from Anzai flows and their K-groups, arXiv preprint arXiv:math/0311425 [math.OA], 2003.

K. Reihani, K-theory of Furstenberg transformation group C^*-algebras, arXiv preprint arXiv:1109.4473 [math.OA], 2011.

Formula

a(n) = constant term of prod(i=1, n, 1+t^(i-.5(n+1))) for odd n and a(n) = constant term of (1+t^(.5))*prod(i=1, n, 1+t^(i-.5(n+1))) for even n.

Sums of antidiagonals of A067059, i.e. a(n) is sum over k of number of partitions of [k(n-k)/2] into up to k parts each no more than n-k. Close to 2^(n+1)*sqrt(6/(Pi*n^3)) and seems to be even closer to something like 2^(n+1)*sqrt(6/(Pi*(n^3+0.9*n^2-0.1825*n+1.5))). - Henry Bottomley, Jul 20 2003

Maple

A084239 := proc(n)
    local tt, c ;
    if type(n, 'odd') then
        product( 1+t^(i-(n+1)/2), i=1..n) ;
    else
        (1+t^(1/2))*product( 1+t^(i-(n+1)/2), i=1..n) ;
    end if;
    tt := expand(%) ;
    for c in tt do
        if c = lcoeff(c) then
            return c ;
        end if;
    end do:
end proc: # R. J. Mathar, Nov 13 2016

Mathematica

a[n_] := SeriesCoefficient[If[OddQ[n], 1, 1 + Sqrt[t]]*Product[1 + t^(i - (n + 1)/2), {i, n}], {t, 0, 0}];
Array[a, 36, 0] (* Jean-François Alcover, Nov 24 2017 *)

Crossrefs

Cf. A000980.

Sequence in context: A000029 A155051 A018137 * A283022 A219186 A049708

Adjacent sequences: A084236 A084237 A084238 * A084240 A084241 A084242

Keyword

nonn

Author

Kamran Reihani (reyhan_k(AT)modares.ac.ir), Jun 21 2003

Extensions

More terms from Henry Bottomley, Jul 20 2003

Status

approved