A225912 Expansion of q * (phi(-q^2) * psi(-q)^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions.

0, 1, -8, 20, 0, -74, 96, -24, 0, 157, -432, 124, 0, 478, 704, -1480, 0, -1198, 792, 3044, 0, -480, -4320, 184, 0, 2351, 3344, -1720, 0, -3282, 5184, -5728, 0, 2480, -4752, 1776, 0, 10326, -6688, 9560, 0, -8886, -8448, -9188, 0, -11618, 32832, 23664, 0, -16231

Offset: 0

Author

Michael Somos, May 20 2013

Keywords

• sign

Comments

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

Examples

			G.f. = q - 8*q^2 + 20*q^3 - 74*q^5 + 96*q^6 - 24*q^7 + 157*q^9 - 432*q^10 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A225923 (bisection?)

Programs

Mathematica

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

PARI

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

Formula

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

Euler transform of period 4 sequence [-8, -8, -8, -12, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^14 (t/i)^6 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A225872.

G.f.: x * (Product_{k>0} (1 - x^k)^2 * (1 - x^(4*k)))^4.