A246836 Expansion of phi(x) * psi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

1, 2, -2, -4, 3, 2, -6, -4, 4, 6, -4, -4, 7, 8, -2, -8, 8, 4, -10, -4, 4, 10, -10, -8, 9, 4, -6, -12, 8, 6, -10, -12, 4, 14, -8, -4, 16, 10, -8, -8, 9, 10, -12, -12, 8, 12, -12, -4, 20, 10, -6, -20, 8, 6, -10, -12, 8, 20, -18, -8, 11, 12, -12, -16, 8, 6, -20

Offset

0,2

Comments

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

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

Formula

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

Euler transform of period 8 sequence [ 2, -5, 2, -1, 2, -5, 2, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 32 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246835.

a(n) = (-1)^floor(n/2) * A045828(n). a(n) = (-1)^n * A246815(n).

a(2*n) = A246835(n). a(2*n + 1) = 2 * A246833(n).

Example

G.f. = 1 + 2*x - 2*x^2 - 4*x^3 + 3*x^4 + 2*x^5 - 6*x^6 - 4*x^7 + 4*x^8 + ...
G.f. = q + 2*q^3 - 2*q^5 - 4*q^7 + 3*q^9 + 2*q^11 - 6*q^13 - 4*q^15 + 4*q^17 + ...

Mathematica

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

Prog

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

Crossrefs

Cf. A045828, A246833, A246835.

Sequence in context: A333920 A356164 A246815 * A246953 A045828 A058526

Adjacent sequences: A246833 A246834 A246835 * A246837 A246838 A246839

Keyword

sign

Author

Michael Somos, Sep 04 2014

Status

approved