A246712 Expansion of chi(x^2) / phi(x) in powers of x where phi(), chi() are Ramanujan theta functions.

1, -2, 5, -10, 18, -32, 55, -90, 145, -228, 351, -532, 795, -1170, 1703, -2452, 3494, -4934, 6910, -9598, 13238, -18134, 24680, -33390, 44921, -60108, 80029, -106044, 139875, -183706, 240284, -313046, 406319, -525490, 677269, -870010, 1114061, -1422210

Offset: 0

Author

Michael Somos, Sep 02 2014

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 2*x + 5*x^2 - 10*x^3 + 18*x^4 - 32*x^5 + 55*x^6 - 90*x^7 + ...
G.f. = 1/q - 2*q^11 + 5*q^23 - 10*q^35 + 18*q^47 - 32*q^59 + 55*q^71 + ...
		

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. A085261.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^4] / QPochhammer[ -x]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^4] / EllipticTheta[ 3, 0, x], {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^4 / (eta(x^8 + A) * eta(x^2 + A)^6), n))};

Formula

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

Expansion of q^(1/12) * eta(q)^2 * eta(q^4)^4 / (eta(q^8) * eta(q^2)^6) in powers of q.

Euler transform of period 8 sequence [-2, 4, -2, 0, -2, 4, -2, 1, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 96^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A085261.