A139632 Expansion of chi(q) * chi(-q^5) in powers of q where chi() is a Ramanujan theta function.

1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 3, 4, 3, 2, 4, 5, 4, 4, 5, 6, 6, 5, 6, 8, 7, 6, 8, 11, 10, 8, 11, 13, 11, 10, 13, 16, 15, 14, 17, 20, 18, 17, 20, 24, 23, 21, 25, 31, 29, 26, 32, 37, 34, 32, 39, 44, 42, 41, 47, 54, 52, 49

Offset

0,13

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

Euler transform of period 20 sequence [ 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, 1, 0, 1, -1, 0, 0, 1, -1, 1, 0, ...].

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 A139631.

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

a(n) = (-1)^floor((n + 1)/2) * A145705(n). - Michael Somos, Sep 07 2015

a(2*n) = A139631(n). a(2*n + 1) = A145703(n). - Michael Somos, Sep 07 2015

a(n) ~ exp(Pi*sqrt(n/10)) / (2^(5/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

Example

G.f. = 1 + x + x^3 + x^4 + x^7 + x^8 + x^9 + x^10 + x^11 + 2*x^12 + x^13 + ...
G.f. = 1/q + q^3 + q^11 + q^15 + q^27 + q^31 + q^35 + q^39 + q^43 + 2*q^47 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^5, x^10], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
nmax = 100; CoefficientList[Series[Product[(1 + x^k) / ((1 + x^(2*k)) * (1 + x^(5*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^2 + A)^2 * eta(x^5 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)), n))};

Crossrefs

Cf. A139631, A145703, A145705.

Sequence in context: A117957 A145702 A145704 * A145705 A029339 A029364

Adjacent sequences: A139629 A139630 A139631 * A139633 A139634 A139635

Keyword

nonn

Author

Michael Somos, Apr 27 2008

Status

approved