A244276 Expansion of q^(-1/4) * eta(q)^8 * eta(q^4)^2 / eta(q^2)^5 in powers of q.

1, -8, 25, -40, 48, -80, 121, -120, 144, -200, 192, -248, 337, -280, 336, -440, 384, -480, 528, -480, 673, -720, 624, -720, 816, -760, 864, -1080, 864, -1000, 1321, -1008, 1200, -1360, 1152, -1440, 1536, -1400, 1488, -1720, 1536, -1760, 2185, -1560, 1872

Offset: 0

Author

Michael Somos, Sep 01 2014

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 8*x + 25*x^2 - 40*x^3 + 48*x^4 - 80*x^5 + 121*x^6 - 120*x^7 + ...
G.f. = q - 8*q^5 + 25*q^9 - 40*q^13 + 48*q^17 - 80*q^21 + 121*q^25 + ...
		

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. A134415.

Programs

Magma

A := Basis( ModularForms( Gamma0(16), 5/2), 180); A[2] - 8*A[6];

Mathematica

a[ n_] := SeriesCoefficient[QPochhammer[ x]^6/EllipticTheta[ 3, 0, x], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x]^4 EllipticTheta[ 2, 0, x]/(2 x^(1/4)), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x]^3 EllipticTheta[ 2, Pi/4, x^(1/2)]^2/(2 x^(1/4)), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(1/2)]^8 / (2 x^(1/4) EllipticTheta[ 2, 0, x]^3 ), {x, 0, n}];

PARI

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

Formula

Expansion of phi(-x)^4 * psi(x^2) = phi(-x)^3 * psi(-x)^2 = f(-x)^6 / phi(x) = psi(-x)^8 / psi(x^2)^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Euler transform of period 4 sequence [ -8, -3, -8, -5, ...].

G.f.: Product_{k>0} (1 - x^k)^5 * (1 + x^(2*k))^2 / (1 + x^k)^3.

Convolution inverse of A134415.