A145707 Expansion of chi(-q) / chi(-q^10) in powers of q where chi() is a Ramanujan theta function.

1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 3, -3, 3, -4, 4, -5, 6, -6, 7, -8, 10, -11, 11, -13, 15, -17, 18, -20, 23, -25, 29, -32, 34, -39, 42, -47, 52, -56, 62, -68, 77, -83, 89, -99, 108, -119, 129, -139, 154, -167, 183, -199, 214, -234, 253, -276, 299, -322, 350

Offset: 0

Author

Michael Somos, Oct 17 2008

Keywords

• sign

Comments

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

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd. - Vaclav Kotesovec, Aug 31 2015

Examples

			G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 + 3*x^10 + ...
G.f. = q^3 - q^11 - q^27 + q^35 - q^43 + q^51 - q^59 + 2*q^67 - 2*q^75 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A145703, A145704, A145705.

Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261735 (m=8), A261733 (m=9).

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^(10*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^10, x^10], {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)

PARI

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

Formula

Expansion of q^(-3/8) * eta(q) * eta(q^20) / (eta(q^2) * eta(q^10)) in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (1280 t)) = f(t) where q = exp(2 Pi i t).

G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(20*k - 10)).

a(n) = (-1)^n * A145703(n) = A145704(2*n + 1) = - A145705(2*n + 1).

a(n) ~ (-1)^n * exp(Pi*sqrt(n/5)) / (4*5^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015