A084239 - OEIS

%I #22 Nov 02 2023 10:55:50

%S 1,2,3,4,6,8,13,20,32,52,90,152,268,472,845,1520,2766,5044,9277,17112,

%T 31724,59008,110162,206260,387282,729096,1375654,2601640,4929378,

%U 9358944,17797100,33904324,64678112,123580884,236413054,452902072

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

%H N. J. A. Sloane, <a href="/A084239/b084239.txt">Table of n, a(n) for n = 0..500</a>

%H K. Reihani, <a href="https://arxiv.org/abs/math/0311425">C*-algebras from Anzai flows and their K-groups</a>, arXiv preprint arXiv:math/0311425 [math.OA], 2003.

%H K. Reihani, <a href="http://arxiv.org/abs/1109.4473">K-theory of Furstenberg transformation group C^*-algebras</a>, arXiv preprint arXiv:1109.4473 [math.OA], 2011.

%F 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.

%F 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

%p A084239 := proc(n)

%p     local tt,c ;

%p     if type(n,'odd') then

%p         product( 1+t^(i-(n+1)/2),i=1..n) ;

%p     else

%p         (1+t^(1/2))*product( 1+t^(i-(n+1)/2),i=1..n) ;

%p     end if;

%p     tt := expand(%) ;

%p     for c in tt do

%p         if c = lcoeff(c) then

%p             return c ;

%p         end if;

%p     end do:

%p end proc: # _R. J. Mathar_, Nov 13 2016

%t a[n_] := SeriesCoefficient[If[OddQ[n], 1, 1 + Sqrt[t]]*Product[1 + t^(i - (n + 1)/2), {i, n}], {t, 0, 0}];

%t Array[a, 36, 0] (* _Jean-François Alcover_, Nov 24 2017 *)

%Y Cf. A000980.

%K nonn

%O 0,2

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

%E More terms from _Henry Bottomley_, Jul 20 2003