A212768 Expansion of (eta(q) * eta(q^3) * eta(q^8) * eta(q^24))^2 in powers of q.

1, -2, -1, 0, 5, 4, -7, 0, -7, 6, -2, 0, 1, -8, 22, 0, 3, -12, 9, 0, -28, -12, -4, 0, 10, 32, 21, 0, -9, 16, -23, 0, 6, -18, -52, 0, 28, 24, -30, 0, 13, -24, 5, 0, 78, 24, 40, 0, -16, -76, -55, 0, 2, -64, 35, 0, -1, 36, 136, 0, -123, -32, -34, 0, -87, 252, 28

Offset

3,2

Links

G. C. Greubel, Table of n, a(n) for n = 3..2500

Formula

Expansion of b(q) * c(q) * b(q^8) * c(q^8) / 9 in powers of q where b(), c() are cubic AGM theta functions.

Euler transform of period 24 sequence [ -2, -2, -4, -2, -2, -4, -2, -4, -4, -2, -2, -4, -2, -2, -4, -4, -2, -4, -2, -2, -4, -2, -2, -8, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 576 (t/i)^4 f(t) where q = exp(2 Pi i t).

G.f.: x^3 * (Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 - x^(8*k)) * (1 - x^(24*k)))^2.

a(4*n) = -2 * A030209(n). a(4*n + 2) = 0.

Example

x^3 - 2*x^4 - x^5 + 5*x^7 + 4*x^8 - 7*x^9 - 7*x^11 + 6*x^12 - 2*x^13 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]* eta[q^3]*eta[q^8]*eta[q^24])^2, {q, 0, 50}], q] (* G. C. Greubel, Aug 11 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<3, 0, n = n-3; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^8 + A) * eta(x^24 + A))^2, n))}

Crossrefs

Cf. A030209.

Sequence in context: A330396 A182376 A030206 * A133336 A269951 A176056

Adjacent sequences: A212765 A212766 A212767 * A212769 A212770 A212771

Keyword

sign

Author

Michael Somos, May 26 2012

Status

approved