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A123477 Expansion of (1 - b(q)) / 3 in powers of q where b(q) is a cubic AGM theta function. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Christian Kassel, 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.

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).

EXAMPLE

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 + ...

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 *)

PROG

(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))))};

CROSSREFS

Cf. A002324, A005928, A033687, A113063.

Sequence in context: A239393 A256637 A113063 * A035225 A298931 A035219

Adjacent sequences:  A123474 A123475 A123476 * A123478 A123479 A123480

KEYWORD

sign,mult

AUTHOR

Michael Somos, Sep 27 2006

STATUS

approved

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Last modified June 1 11:51 EDT 2020. Contains 334762 sequences. (Running on oeis4.)