A138517 Expansion of (phi(-q^5) / phi(-q))^2 in powers of q where phi() is a Ramanujan theta function.

1, 4, 12, 32, 76, 164, 336, 656, 1228, 2228, 3932, 6768, 11408, 18872, 30688, 49152, 77644, 121096, 186684, 284720, 429916, 643168, 953904, 1403312, 2048784, 2969764, 4275656, 6116480, 8696864, 12294680, 17285776, 24176288, 33645132

Offset: 0

Author

Michael Somos, Mar 23 2008

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

Examples

			1 + 4*q + 12*q^2 + 32*q^3 + 76*q^4 + 164*q^5 + 336*q^6 + 656*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. 4 * A095846(n) = a(n) unless n=0. Convolution inverse of A138518. Convolution square of A138526.

Programs

Mathematica

eta[x_] := x^(1/24)*QPochhammer[x]; A138517[n_] := SeriesCoefficient[ ((eta[q^5]/eta[q])^2*eta[q^2]/eta[q^10])^2, {q, 0, n}]; Table[ A138517[n], {n, 0, 50}] (* G. C. Greubel, Sep 29 2017 *)

PARI

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

Formula

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

Euler transform of period 10 sequence [ 4, 2, 4, 2, 0, 2, 4, 2, 4, 0, ...].

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

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (1 - u) * (1 - 5*u) * v * (1 - v) * (1 - 5*v).

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138516.

G.f.: (Product_{k>0} P(5, x^k) / P(10, x^k))^2 where P(n, x) is the n-th cyclotomic polynomial.

a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(3/4) * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018

Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 3/10 + (1/10)*sqrt(5) + (1/10)*sqrt(10 + 6*sqrt(5)). - Simon Plouffe, Mar 04 2021