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A123477 Expansion of (1 - b(q)) / 3 in powers of q where b(q) is a cubic AGM theta function.

%I A123477 #17 Feb 22 2021 08:12:07
%S A123477 1,0,-2,1,0,0,2,0,-2,0,0,-2,2,0,0,1,0,0,2,0,-4,0,0,0,1,0,-2,2,0,0,2,0,
%T A123477 0,0,0,-2,2,0,-4,0,0,0,2,0,0,0,0,-2,3,0,0,2,0,0,0,0,-4,0,0,0,2,0,-4,1,
%U A123477 0,0,2,0,0,0,0,0,2,0,-2,2,0,0,2,0,-2,0,0,-4,0,0,0,0,0,0,4,0,-4,0,0,0,2,0,0,1,0,0,2,0,0
%N A123477 Expansion of (1 - b(q)) / 3 in powers of q where b(q) is a cubic AGM theta function.
%C A123477 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A123477 Denoted by lambda(n) on page 4 (1.7) in Kassel and Reutenauer arXiv:1610.07793. - _Michael Somos_, Dec 10 2017
%H A123477 Seiichi Manyama, <a href="/A123477/b123477.txt">Table of n, a(n) for n = 1..1000</a>
%H A123477 Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1610.07793">Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv preprint arXiv:1610.07793 [math.NT], 2016.
%F A123477 Moebius transform is period 9 sequence [1, -1, -3, 1, -1, 3, 1, -1, 0, ...].
%F A123477 a(n) is multiplicative and a(p^e) = -2 if p = 3 and e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6).
%F A123477 a(3*n + 2) = 0. a(3*n + 1) = A033687(n), a(3*n) = -2*A002324(n).
%F A123477 -3*a(n) = A005928(n) unless n=0. |a(n)| = A113063(n).
%e A123477 G.f. = q - 2*q^3 + q^4 + 2*q^7 - 2*q^9 - 2*q^12 + 2*q^13 + q^16 + 2*q^19 + ...
%p A123477 A123477 := proc(n)
%p A123477     local a,pe,p,e;
%p A123477     a := 1;
%p A123477     for pe in ifactors(n)[2] do
%p A123477         p := op(1,pe) ;
%p A123477         e := op(2,pe) ;
%p A123477         if modp(p,6) = 1 then
%p A123477             a := a*(e+1) ;
%p A123477         elif modp(p,6) in {2,5} then
%p A123477             a := a*(1+(-1)^e)/2 ;
%p A123477         elif e > 0 then
%p A123477             a := -2*a ;
%p A123477         end if;
%p A123477     end do:
%p A123477     a ;
%p A123477 end proc:
%p A123477 seq(A123477(n),n=1..100) ; # _R. J. Mathar_, Feb 22 2021
%t A123477 a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -1, -3, 1, -1, 3, 1, -1, 0} [[Mod[#, 9, 1]]] &]]; (* _Michael Somos_, Dec 10 2017 *)
%o A123477 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, -3, 1, -1, 3, 1, -1] [d%9+1]))};
%o A123477 (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, -2, p%6==1, e+1, !(e%2))))};
%Y A123477 Cf. A002324, A005928, A033687, A113063.
%K A123477 sign,mult
%O A123477 1,3
%A A123477 _Michael Somos_, Sep 27 2006