A228864 Expansion of 1 + q * (psi(-q^5) / psi(-q))^2 in powers of q where psi() is a Ramanujan theta function.

1, 1, 2, 3, 6, 11, 16, 24, 38, 57, 82, 117, 168, 238, 328, 448, 614, 834, 1114, 1480, 1966, 2592, 3384, 4398, 5704, 7361, 9436, 12045, 15344, 19470, 24576, 30922, 38822, 48576, 60548, 75259, 93342, 115454, 142360, 175104, 214958, 263262, 321584, 391993, 476952

Offset: 0

Author

Michael Somos, Sep 05 2013

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 11*x^5 + 16*x^6 + 24*x^7 + 38*x^8 + ...
		

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. A138520, A210458, A225849.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 1 + (EllipticTheta[ 2, Pi/4, q^(5/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)])^2, {q, 0, n}]; (* Michael Somos, Oct 26 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^5] / EllipticTheta[ 3, 0, q])^2 QPochhammer[ q^5, -q^5] / QPochhammer[ q, -q]^5, {q, 0, n}]; (* Michael Somos, Oct 26 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^10 + A)^8 / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A)^3 * eta(x^20 + A)^3), n))};

Formula

Expansion of (phi(q^5) / phi(q))^2 * (chi^5(q) / chi(q^5)) in powers of q where phi(), chi() are Ramanujan theta functions.

Expansion of eta(q^10)^8 / (eta(q) * eta(q^4) * eta(q^5)^3 * eta(q^20)^3) in powers of q.

Euler transform of period 20 sequence [ 1, 1, 1, 2, 4, 1, 1, 2, 1, -4, 1, 2, 1, 1, 4, 2, 1, 1, 1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A225849.

a(n) = A210458(n) unless n=0. a(n) = (-1)^n * A138520(n).

a(n) ~ exp(2*Pi*sqrt(n/5)) / (2 * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017