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

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, 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, 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

Offset: 1

Author

Michael Somos, Sep 27 2006

Keywords

• sign
• mult

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Denoted by lambda(n) on page 4 (1.7) in Kassel and Reutenauer arXiv:1610.07793. - Michael Somos, Dec 10 2017

Examples

			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 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv preprint arXiv:1610.07793 [math.NT], 2016.

Crossrefs

Cf. A002324, A005928, A033687, A113063.

Programs

Maple

A123477 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if modp(p,6) = 1 then
            a := a*(e+1) ;
        elif modp(p,6) in {2,5} then
            a := a*(1+(-1)^e)/2 ;
        elif e > 0 then
            a := -2*a ;
        end if;
    end do:
    a ;
end proc:
seq(A123477(n),n=1..100) ; # R. J. Mathar, Feb 22 2021

Mathematica

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 *)

PARI

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, -3, 1, -1, 3, 1, -1] [d%9+1]))};

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))))};

Formula

Moebius transform is period 9 sequence [1, -1, -3, 1, -1, 3, 1, -1, 0, ...].

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).

a(3*n + 2) = 0. a(3*n + 1) = A033687(n), a(3*n) = -2*A002324(n).

-3*a(n) = A005928(n) unless n=0. |a(n)| = A113063(n).