A132965 Expansion of f(-q^8) * chi(q)^2 in powers of q where f(), chi() are Ramanujan theta functions.

1, 2, 1, 2, 4, 4, 5, 6, 8, 10, 12, 14, 17, 22, 24, 30, 36, 40, 48, 56, 65, 76, 88, 100, 116, 134, 152, 174, 200, 226, 257, 292, 328, 372, 420, 472, 532, 598, 668, 750, 840, 936, 1045, 1166, 1296, 1442, 1604, 1776, 1972, 2186, 2416, 2672, 2952, 3256, 3592, 3960

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 8^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132966.

G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k)) * (1 + x^(4*k)) / (1 + x^(2*k)).

a(n) ~ exp(sqrt(n)*Pi/2) / (4*sqrt(n)). - Vaclav Kotesovec, Sep 08 2015

Example

G.f. = 1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 8*x^8 + 10*x^9 + ...
G.f. = q + 2*q^5 + q^9 + 2*q^13 + 4*q^17 + 4*q^21 + 5*q^25 + 6*q^29 + 8*q^33 + ...

Mathematica

nmax = 60; CoefficientList[Series[Product[(1 + x^k)^2 * (1 - x^(2*k)) * (1 + x^(4*k)) / (1 + x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^8] QPochhammer[ -x, x^2]^2, {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)

Prog

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

Crossrefs

Cf. A132966.

Sequence in context: A153898 A108802 A023673 * A365005 A022597 A073252

Adjacent sequences: A132962 A132963 A132964 * A132966 A132967 A132968

Keyword

nonn

Author

Michael Somos, Aug 23 2007

Status

approved