A246811 Expansion of phi(x)^2 * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 4, 4, 0, 5, 12, 4, 0, 8, 12, 8, 0, 5, 16, 12, 0, 8, 24, 4, 0, 16, 12, 12, 0, 9, 24, 12, 0, 8, 36, 12, 0, 16, 12, 16, 0, 8, 28, 16, 0, 17, 36, 8, 0, 24, 24, 8, 0, 8, 36, 28, 0, 16, 36, 12, 0, 16, 24, 20, 0, 13, 24, 24, 0, 24, 60, 8, 0, 16, 36, 16, 0, 16, 28

Offset: 0

Author

Michael Somos, Sep 03 2014

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 4*x + 4*x^2 + 5*x^4 + 12*x^5 + 4*x^6 + 8*x^8 + 12*x^9 + ...
G.f. = q + 4*q^3 + 4*q^5 + 5*q^9 + 12*q^11 + 4*q^13 + 8*q^17 + 12*q^19 + ...
		

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. A008443, A033717, A045831, A045834, A213022.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 EllipticTheta[ 2, 0, x^2] / (2 x^(1/2)), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)]^4 / (16 x^(1/2) EllipticTheta[ 3, 0, x^2]), {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^8 + A)^2 / (eta(x + A)^4 * eta(x^4 + A)^5), n))};

Formula

Expansion of psi(x)^4 / phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

Expansion of q^(-1/2) * eta(q^2)^10 * eta(q^8)^2 / (eta(q)^4 * eta(q^4)^5) in powers of q.

Euler transform of period 8 sequence [4, -6, 4, -1, 4, -6, 4, -3, ...].

2 * a(n) = A033717(4*n + 2). a(2*n) = A045834(n). a(4*n) = A213022(n). a(4*n + 1) = 4 * A008443(n). a(4*n + 2) = 4 * A045831(n). a(4*n + 3) = 0.