A227239 Expansion of q * f(-q^2)^12 + 8 * q^2 * f(-q^4)^12 in powers of q where f() is a Ramanujan theta function.

1, 8, -12, 0, 54, -96, -88, 0, -99, 432, 540, 0, -418, -704, -648, 0, 594, -792, 836, 0, 1056, 4320, -4104, 0, -209, -3344, 4104, 0, -594, -5184, 4256, 0, -6480, 4752, -4752, 0, -298, 6688, 5016, 0, 17226, 8448, -12100, 0, -5346, -32832, -1296, 0, -9063

Offset: 1

Author

Michael Somos, Sep 02 2013

Keywords

• sign
• mult

Comments

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

The bisection {a(2*n+1)} is Glaisher's Omega function, A000735. The other bisection, {a(2*n)}, begins 8, 0, -96, 0, 432, 0, -704, 0, -792, 0, 4320, 0, -3344, ..., and if this in turn is bisected and then divided by 8, we again obtain A000735. - N. J. A. Sloane, Nov 25 2018

Examples

			q + 8*q^2 - 12*q^3 + 54*q^5 - 96*q^6 - 88*q^7 - 99*q^9 + 432*q^10 + ...
		

Links

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

Crossrefs

Cf. A000735.

Programs

Magma

A := Basis( CuspidalSubspace( ModularForms( Gamma1(8), 6))); PowerSeries( A[1] +8*A[2] -12*A[3] +54*A[5] -96*A[6] -88*A[7], 50);

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^12 + 8 q^2 QPochhammer[ q^4]^12, {q, 0, n}]

PARI

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^12 + 8 * x * eta(x^4 + A)^12, n))}

Formula

Expansion of eta(q^2)^12 + 8 * eta(q^4)^12 in powers of q.

a(n) is multiplicative with a(2) = 8, a(2^e) = 0 if e > 1, a(p^e) = a(p) * a(p^(e-1)) - p^5 * a(p^(e-2)) if p > 2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8^3 (t / i)^6 f(t) where q = exp(2 Pi i t).

a(4*n) = 0. a(2*n + 1) = A000735(n). a(4*n + 2) = 8 * A000735(n).