A226010 Expansion of u1 * (u1 + u4) * (u1^2 - 2 * u1 * u4 - u4^2) / eta(q^2)^4 in powers of q where u1 = eta(q)^8 and u4 = 32 * eta(q^4)^8.

1, -64, -1836, 4096, 3990, 117504, -433432, -262144, 1776573, -255360, 1619772, -7520256, -10878466, 27739648, -7325640, 16777216, 60569298, -113700672, -243131740, 16343040, 795781152, -103665408, -606096456, 481296384, -1204783025, 696221824, -334611000

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 - 1836*q^3 + 4096*q^4 + 3990*q^5 + 117504*q^6 - 433432*q^7 + ...

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q];  A := eta[q]^8; B := 32*eta[q^4]^8; Drop[CoefficientList[Series[A*(A + B)*(A^2 - 2*A*B - B^2)/eta[q^2]^4, {q, 0, 50}], q], 1] (* G. C. Greubel, Aug 09 2018 *)

Prog

(PARI) {a(n) = my(A, u1, u4); if( n<1, 0, n--; A = x * O(x^n); u1 = eta(x + A)^8; u4 = 32 * x * eta(x^4 + A)^8; polcoeff( u1 * (u1 + u4) * (u1^2 - 2* u1 * u4 - u4^2) / eta(x^2 + A)^4, n))};
(Sage) A = CuspForms( Gamma1(2), 14, prec=28) . basis(); A[0] - 64*A[1];
(Magma) A := Basis( CuspForms( Gamma1(2), 14), 28); A[1] - 64*A[2];

Crossrefs

Cf. A226086.

Sequence in context: A065788 A264275 A283689 * A203539 A116133 A010841

Adjacent sequences: A226007 A226008 A226009 * A226011 A226012 A226013

Keyword

sign,mult

Author

Michael Somos, May 22 2013

Status

approved