A262160 Expansion of psi(x^6) / psi(x) in powers of x where psi() is a Ramanujan theta function.

1, -1, 1, -2, 3, -4, 6, -8, 11, -15, 19, -25, 33, -42, 53, -68, 86, -107, 134, -166, 205, -253, 309, -377, 460, -557, 672, -811, 974, -1166, 1394, -1661, 1975, -2344, 2773, -3275, 3863, -4543, 5333, -6253, 7316, -8544, 9964, -11600, 13484, -15653, 18140

Offset

0,4

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^(-5/8) * eta(q) * eta(q^12)^2 / (eta(q^2)^2 * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [ -1, 1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 0, ...].

a(n) = (-1)^n * A132217(n).

Product_{k>0} (1 - x^(12*k)) * (1 - x^(2*k) + x^(4*k)) / (1 - (-x)^k). - Michael Somos, Oct 04 2015

Example

G.f. = 1 - x + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 6*x^6 - 8*x^7 + 11*x^8 + ...
G.f. = q^5 - q^13 + q^21 - 2*q^29 + 3*q^37 - 4*q^45 + 6*q^53 - 8*q^61 + ...

Mathematica

a[ n_] := SeriesCoefficient[ x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};

Crossrefs

Cf. A132217.

Sequence in context: A117995 A033834 A127419 * A132217 A265254 A303944

Adjacent sequences: A262157 A262158 A262159 * A262161 A262162 A262163

Keyword

sign

Author

Michael Somos, Sep 13 2015

Status

approved