A227317 Expansion of psi(x)^6 * phi(-x)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

1, 2, -5, -10, 5, 6, 10, 40, -20, -50, 19, -52, -30, 50, -25, 74, 97, 50, -25, -140, 69, -34, -100, -50, -185, -6, 83, 310, -60, -60, 410, -128, 145, -100, -245, 250, -87, -90, -400, -410, -151, 362, 185, -50, 285, 30, 150, -240, 500, 370, -68, 222, 5, -190

Offset: 0

Author

Michael Somos, Sep 02 2013

Keywords

• sign

Comments

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

Examples

			1 + 2*x - 5*x^2 - 10*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 40*x^7 - 20*x^8 + ...
q^3 + 2*q^7 - 5*q^11 - 10*q^15 + 5*q^19 + 6*q^23 + 10*q^27 + 40*q^31 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A008439, A010816, A215600, A227695.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^5 / QPochhammer[ q])^2, {q, 0, n}]

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / eta(x + A))^2, n))}

Formula

Expansion of psi(x)^5 * f(-x)^3 = psi(x)^2 * f(-x^2)^6 in powers of x where psi(), f() are Ramanujan theta functions.

Expansion of q^(-3/4) * (eta(q^2)^5 / eta(x))^2 in powers of q.

Euler transform of period 2 sequence [ 2, -8, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 128 (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227695.

G.f.: (Product_{k>0} (1 - x^(2*k))^5 / (1 - x^k))^2.

Convolution of A008439 and A010816.

-8 * a(n) = A215600(2*n + 1).