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

0,2

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 (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

Example

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

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 *)

Prog

(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))};

Crossrefs

Cf. A132980.

Sequence in context: A262295 A034182 A348885 * A022570 A152881 A000333

Adjacent sequences: A132982 A132983 A132984 * A132986 A132987 A132988

Keyword

nonn

Author

Michael Somos, Sep 07 2007

Extensions

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

Status

approved