A216711 Expansion of q * (phi(q) * psi(-q))^8 in powers of q where phi(), psi() are Ramanujan theta functions.

1, 8, 12, -64, -210, 96, 1016, 512, -2043, -1680, 1092, -768, 1382, 8128, -2520, -4096, 14706, -16344, -39940, 13440, 12192, 8736, 68712, 6144, -34025, 11056, -50760, -65024, -102570, -20160, 227552, 32768, 13104, 117648, -213360, 130752, 160526, -319520

Offset: 1

Author

Michael Somos, Apr 10 2013

Keywords

• sign
• mult
• look

Comments

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

Examples

			G.f. = q + 8*q^2 + 12*q^3 - 64*q^4 - 210*q^5 + 96*q^6 + 1016*q^7 + 512*q^8 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A002288, A134461.

Programs

Magma

A := Basis( CuspForms( Gamma0(4), 8), 39); A[1] + 8*A[2]; /* Michael Somos, Jun 10 2015 */

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(1/2)] / 2)^8, {q, 0, n}];

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^4 / eta(x + A) / eta(x^4 + A) )^8, n))};

Formula

Expansion of (eta(q^2) / (eta(q) * eta(q^4)))^8 in powers of q.

a(n) is multiplicative with a(2) = 8, a(2^e) = -(-8)^e if e>1, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)) if p>2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 256 (t/i)^8 f(t) where q = exp(2 Pi i t).

a(n) = -(-1)^n * A002288(n). Convolution square of A134461.