A181905 Expansion of (b(q^3)^3 - b(q)^3) / 9 in powers of q where b() is a cubic AGM theta function.

1, -3, 0, 13, -24, 0, 50, -51, 0, 72, -120, 0, 170, -150, 0, 205, -288, 0, 362, -312, 0, 360, -528, 0, 601, -510, 0, 650, -840, 0, 962, -819, 0, 864, -1200, 0, 1370, -1086, 0, 1224, -1680, 0, 1850, -1560, 0, 1584, -2208, 0, 2451, -1803, 0, 2210, -2808, 0

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Formula

Expansion of a(q^3) * b(q) * c(q^3) / 3 in powers of q where a(), b(), c() are cubic AGM theta functions.

Expansion of ((eta(q^3)^3 / eta(q^9))^3 - (eta(q)^3 / eta(q^3))^3) / 9 in powers of q.

a(3*n) = 0.

Conjecture: multiplicative with a(3^e) = 0, a(p^e) = ((p^2)^(e+1)-1)/(p^2-1) for p == 1 (mod 3), a(p^e) = (1-(-p^2)^(e+1))/(p^2+1) for p == 2 (mod 3). - Andrew Howroyd, Aug 05 2018

Example

q - 3*q^2 + 13*q^4 - 24*q^5 + 50*q^7 - 51*q^8 + 72*q^10 - 120*q^11 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; Rest[CoefficientList[Series[ ((eta[q^3]^3/ eta[q^9])^3 - (eta[q]^3/eta[q^3])^3)/9, {q, 0, 50}], q]] (* G. C. Greubel, Aug 11 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, A = x * O(x^n); polcoeff( ((eta(x^3 + A)^3 / eta(x^9 + A))^3 - (eta(x + A)^3 / eta(x^3 + A))^3) / 9, n))}

Crossrefs

Sequence in context: A058896 A186748 A222754 * A350826 A008403 A138349

Adjacent sequences: A181902 A181903 A181904 * A181906 A181907 A181908

Keyword

sign

Author

Michael Somos, Apr 01 2012

Status

approved