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

%I A084239 #22 Nov 02 2023 10:55:50
%S A084239 1,2,3,4,6,8,13,20,32,52,90,152,268,472,845,1520,2766,5044,9277,17112,
%T A084239 31724,59008,110162,206260,387282,729096,1375654,2601640,4929378,
%U A084239 9358944,17797100,33904324,64678112,123580884,236413054,452902072
%N A084239 Rank of K-groups of Furstenberg transformation group C*-algebras of n-torus.
%H A084239 N. J. A. Sloane, <a href="/A084239/b084239.txt">Table of n, a(n) for n = 0..500</a>
%H A084239 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 A084239 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 A084239 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 A084239 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 A084239 := proc(n)
%p A084239     local tt,c ;
%p A084239     if type(n,'odd') then
%p A084239         product( 1+t^(i-(n+1)/2),i=1..n) ;
%p A084239     else
%p A084239         (1+t^(1/2))*product( 1+t^(i-(n+1)/2),i=1..n) ;
%p A084239     end if;
%p A084239     tt := expand(%) ;
%p A084239     for c in tt do
%p A084239         if c = lcoeff(c) then
%p A084239             return c ;
%p A084239         end if;
%p A084239     end do:
%p A084239 end proc: # _R. J. Mathar_, Nov 13 2016
%t A084239 a[n_] := SeriesCoefficient[If[OddQ[n], 1, 1 + Sqrt[t]]*Product[1 + t^(i - (n + 1)/2), {i, n}], {t, 0, 0}];
%t A084239 Array[a, 36, 0] (* _Jean-François Alcover_, Nov 24 2017 *)
%Y A084239 Cf. A000980.
%K A084239 nonn
%O A084239 0,2
%A A084239 Kamran Reihani (reyhan_k(AT)modares.ac.ir), Jun 21 2003
%E A084239 More terms from _Henry Bottomley_, Jul 20 2003