A113297 Expansion of chi(-q) / chi(-q^7) in powers of q where chi() is a Ramanujan theta function.

1, -1, 0, -1, 1, -1, 1, 0, 1, -2, 1, -1, 2, -2, 3, -3, 3, -4, 4, -4, 5, -4, 4, -6, 6, -7, 7, -8, 11, -11, 10, -12, 14, -15, 15, -14, 17, -20, 19, -21, 24, -26, 30, -31, 32, -37, 38, -40, 45, -44, 47, -54, 56, -60, 64, -68, 79, -83, 83, -92, 100, -105, 110, -112, 123, -136, 138, -147, 160, -170, 185, -194, 203

Offset: 0

Author

Michael Somos, Oct 23 2005

Keywords

• sign

Comments

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

Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

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^8 - 2*x^9 + x^10 - x^11 + ...
G.f. = q - q^5 - q^13 + q^17 - q^21 + q^25 + q^33 - 2*q^37 + q^41 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
• F. G. Garvan and H. Yesilyurt, Shifted and shiftless partition identities II, arXiv:math/0605317 [math.NT], 2003.
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 14.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A097793.

Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A133563 (m=5), A261736 (m=6), A261735 (m=8), A261733 (m=9), A145707 (m=10).

Programs

Maple

seq(coeff(series(mul((1+x^(7*k))/(1+x^k),k=1..n), x,n+1),x,n),n=0..80); # Muniru A Asiru, Jul 29 2018

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^14] / (QPochhammer[ x^2] QPochhammer[ x^7]), {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x  + A) * eta(x^14 + A) / (eta(x^2 + A) * eta(x^7 + A)), n))};

Formula

Expansion of q^(-1/4) * eta(q) * eta(q^14) / ( eta(q^2) * eta(q^7) ) in powers of q.

Euler transform of period 14 sequence [ -1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, ...].

G.f. A(x) = G(x^7) * H(x^2) - x * G(x^2) * H(x^7) where G(x) and H(x) are the Rogers-Ramanujan functions.

G.f.: Product_{k>0} (1 + x^(7*k)) / (1 + x^k).

Expansion of chi(-q) / chi(-q^7) in powers of q where chi() is a Ramanujan theta function.

G.f. is a period 1 Fourier series which satisfies f(-1 / (224 t)) = f(t) where q = exp(2 Pi i t).

G.f.: Product_{k>0} P14(x^k) where P14 is the 14th cyclotomic polynomial.

Convolution inverse is A097793.

a(n) ~ (-1)^n * exp(Pi*sqrt(n/7)) / (2^(3/2) * 7^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 31 2015