A226086 Expansion of (2 * eta(q^2)^24 - eta(q)^16 * eta(q^4)^8)^3 / (eta(q)^4 * eta(q^2) * eta(q^4)^6)^4 in powers of q.

1, 64, 1236, 4096, -57450, 79104, 64232, 262144, -66627, -3676800, 2464572, 5062656, 8032766, 4110848, -71008200, 16777216, 71112402, -4264128, 136337060, -235315200, 79390752, 157732608, -1186563144, 324009984, 2079799375, 514097024, -2052934200, 263094272

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..2500

Formula

a(n) is multiplicative with a(2^n) = 64^n, a(p^e) = a(p) * a(p^(e-1)) - p^13 * a(p^(e-2)) if p>2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 128 (t/i)^14 f(t) where q = exp(2 Pi i t).

a(2*n) = 64 * a(n).

Example

G.f. = q + 64*q^2 + 1236*q^3 + 4096*q^4 - 57450*q^5 + 79104*q^6 + 64232*q^7 + ...

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; Drop[CoefficientList[Series[(2* eta[q^2]^24 - eta[q]^16*eta[q^4]^8)^3/(eta[q]^4*eta[q^2]*eta[q^4]^6)^4, {q, 0, 50}], q], 1] (* G. C. Greubel, Aug 09 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (2 * eta(x^2 + A)^24 - eta(x + A)^16 * eta(x^4 + A)^8)^3 / (eta(x + A)^4 * eta(x^2 + A) * eta(x^4 + A)^6)^4, n))};
(Sage) A = CuspForms( Gamma1(2), 14, prec=29) . basis(); A[0] + 64*A[1];
(Magma) A := Basis( CuspForms( Gamma1(2), 14), 29); A[1] + 64*A[2];

Crossrefs

Sequence in context: A128987 A240788 A240294 * A017031 A333812 A283812

Adjacent sequences: A226083 A226084 A226085 * A226087 A226088 A226089

Keyword

sign,mult

Author

Michael Somos, May 25 2013

Status

approved