A132985 Expansion of chi(-q^5) / chi(-q)^5 in powers of q where chi() is a Ramanujan theta function.

1, 5, 15, 40, 95, 205, 420, 820, 1535, 2785, 4915, 8460, 14260, 23590, 38360, 61440, 97055, 151370, 233355, 355900, 537395, 803960, 1192380, 1754140, 2560980, 3712205, 5344570, 7645600, 10871080, 15368350, 21607220, 30220360, 42056415, 58249680, 80310510

Offset: 0

Author

Michael Somos, Sep 07 2007

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 5*q + 15*q^2 + 40*q^3 + 95*q^4 + 205*q^5 + 420*q^6 + 820*q^7 + ...
		

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

Crossrefs

Cf. A132980.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[(1 + x^k)^5 / (1 + x^(5*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^5, q^10] / QPochhammer[ q, q^2]^5, {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)

PARI

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

Formula

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

Euler transform of period 10 sequence [ 5, 0, 5, 0, 4, 0, 5, 0, 5, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 + v + u * v * (2 - 4 * v).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * v * (1 + 3 * u - 4 * u^2) * (1 + 3 * v - 4 * v^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132980.

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

a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(11/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

Extensions

Typo in a(32) corrected by G. C. Greubel, Sep 28 2017