A106508 Expansion of psi(x)^4 * chi(-x^2)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

1, 4, 4, 0, 2, 0, -8, 0, -5, -16, 4, 0, -10, 0, -8, 0, 9, 8, 0, 0, 14, 0, 16, 0, -10, 32, 4, 0, 0, 0, 8, 0, 14, -20, -20, 0, 2, 0, 0, 0, -11, -16, -20, 0, -32, 0, 16, 0, 0, -40, 4, 0, 14, 0, -8, 0, -9, 32, -20, 0, 26, 0, 0, 0, 2, 36, 28, 0, 0, 0, 16, 0, 16, 0, 28, 0, -22, 0, 0, 0, 14, 56, -16, 0, 0, 0, -40, 0, 0

Offset: 0

Author

Michael Somos, May 24 2005

Keywords

• sign

Comments

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

Examples

			1 + 4*x + 4*x^2 + 2*x^4 - 8*x^6 - 5*x^8 - 16*x^9 + 4*x^10 - 10*x^12 + ...
q + 4*q^4 + 4*q^7 + 2*q^13 - 8*q^19 - 5*q^25 - 16*q^28 + 4*q^31 - 10*q^37 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• C. Adiga, N. Anitha and T. Kim, Transformations of Ramanujan's Summation Formula and its Applications, arXiv:math/0501528 [math.NT], 2005. See page 5.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A187149.

Programs

Mathematica

a[n_]:= SeriesCoefficient[(x^(-1/2)/16)*EllipticTheta[2, 0, x^(1/2)]^4* QPochhammer[x^2, x^4]^2, {x, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jan 04 2018 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 / (eta(x^4 + A)^2 * eta(x + A)^4), n))}

Formula

Expansion of q^(-1/3) * eta(q^2)^10 / (eta(q)^4 * eta(q^4)^2) in powers of q.

Euler transform of period 4 sequence [4, -6, 4, -4, ...].

a(n) = (-1)^n * A187149(n). a(4*n + 3) = a(8*n + 5) = 0.

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