A258096 Expansion of psi(x^4) * phi(-x^4)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta function.

1, 2, 4, 8, 7, 10, 12, 8, 18, 18, 16, 24, 21, 20, 28, 32, 20, 32, 36, 24, 42, 42, 28, 48, 57, 36, 52, 40, 36, 58, 60, 56, 48, 66, 48, 72, 74, 42, 80, 80, 61, 82, 72, 56, 90, 96, 64, 72, 98, 70, 100, 104, 64, 106, 108, 72, 114, 96, 84, 144, 111, 84, 104, 128

Offset: 0

Author

Michael Somos, May 19 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 7*x^4 + 10*x^5 + 12*x^6 + 8*x^7 + ...
G.f. = q + 2*q^3 + 4*q^5 + 8*q^7 + 7*q^9 + 10*q^11 + 12*q^13 + 8*q^15 + ...
		

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. A113417, A113419, A209940.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4]^7 / (QPochhammer[ x]^2 QPochhammer[ x^8]^2), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^2] EllipticTheta[ 4, 0, x^4]^4 / (EllipticTheta[ 4, 0, x] 2 x^(1/2)), {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^7 / (eta(x + A)^2 * eta(x^8 + A)^2), n))};

Formula

Expansion of q^(-1/2) * eta(q^2) * eta(q^4)^7 / (eta(q)^2 * eta(q^8)^2) in powers of q.

Euler transform of period 8 sequence [ 2, 1, 2, -6, 2, 1, 2, -4, ...].

a(n) = (-1)^n * A209940(n) = (-1)^floor(n/2) * A113419(n) = (-1)^(n + floor(n/2)) * A113417(n).