A139631 Expansion of chi(x^5) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function.

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 3, 2, 4, 2, 5, 4, 6, 5, 8, 6, 11, 8, 13, 10, 16, 14, 20, 17, 24, 21, 31, 26, 37, 32, 44, 41, 54, 49, 64, 59, 79, 72, 94, 86, 111, 106, 132, 126, 156, 149, 187, 178, 219, 210, 257, 251, 302, 295, 352, 346, 416, 406, 483, 474, 560

Offset

0,7

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^(1/8) * eta(q^4) * eta(q^10)^2 / (eta(q^2) * eta(q^5) * eta(q^20)) in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139632.

G.f.: Product_{k>0} (1 + x^(2*k)) * (1 + x^(5*k)) / (1 + x^(10*k)).

a(n) = A139632(2*n).

a(n) ~ exp(Pi*sqrt(n/5)) / (4 * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

Example

G.f. = 1 + x^2 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + 3*x^10 + 2*x^11 + ...
G.f. = 1/q + q^15 + q^31 + q^39 + 2*q^47 + q^55 + 2*q^63 + q^71 + 3*q^79 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^2] QPochhammer[ -x^5, x^10], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k)) * (1 + x^(5*k)) / (1 + x^(10*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^10 + A)^2 / (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};

Crossrefs

Cf. A139632.

Sequence in context: A025806 A025802 A145706 * A029177 A321298 A261554

Adjacent sequences: A139628 A139629 A139630 * A139632 A139633 A139634

Keyword

nonn

Author

Michael Somos, Apr 27 2008

Status

approved