A173763 Expansion of (eta(q^2)^7 / eta(q^4)^2)^4 + 16 * (eta(q)^2 * eta(q^2) * eta(q^4)^2)^4 in powers of q.

1, 16, -156, 256, 870, -2496, -952, 4096, 4653, 13920, -56148, -39936, 178094, -15232, -135720, 65536, -247662, 74448, 315380, 222720, 148512, -898368, 204504, -638976, -1196225, 2849504, 2344680, -243712, -3840450, -2171520, -1309408, 1048576, 8759088, -3962592, -828240, 1191168, 4307078

Offset: 1

Author

Michael Somos, Feb 23 2010

Keywords

• sign
• mult

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = q + 16*q^2 - 156*q^3 + 256*q^4 + 870*q^5 - 2496*q^6 - 952*q^7 + 4096*q^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• LMFDB, Newform orbit 2.10.a.a.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A002288.

Programs

Magma

Basis( CuspForms( Gamma1(2), 10), 50) [1]; /* Michael Somos, May 27 2014 */

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2]^7 / QPochhammer[ q^4]^2)^4 + 16 q^2 (QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^2)^4, {q, 0, n}]; (* Michael Somos, May 28 2013 *)

PARI

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / eta(x^4 + A)^2)^4 + 16 * x * (eta(x + A)^2 * eta(x^2 + A) * eta(x^4 + A)^2)^4, n))};

PARI

q='q+O('q^99); Vec((eta(q^2)^7/eta(q^4)^2)^4+16*q*(eta(q)^2*eta(q^2)*eta(q^4)^2)^4) \\ Altug Alkan, Apr 18 2018

Sage

CuspForms( Gamma1(2), 10, prec=50).0; # Michael Somos, May 28 2013

Formula

Expansion of q * (psi(q)^3 * phi(-q)^2)^4 * ((phi(q) / psi(q))^4 + 16 * q * (psi(q) / phi(q))^4) in powers of q where phi(), psi() are Ramanujan theta functions.

a(n) is multiplicative with a(2^e) = 16^e, a(p^e) = a(p) * a(p^(e-1)) - p^9 * a(p^(e-2)) if p>2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 32 (t / i)^10 f(t) where q = exp(2 Pi i t).