A215348 Expansion of q * phi(q) * psi(q^8) / (phi(-q) * phi(q^4)) in powers of q where phi(), psi() are Ramanujan theta functions.

1, 4, 8, 16, 30, 48, 80, 128, 197, 312, 472, 704, 1046, 1504, 2160, 3072, 4306, 6036, 8360, 11488, 15712, 21264, 28656, 38400, 51127, 67864, 89552, 117632, 153926, 200352, 259904, 335872, 432336, 554952, 709728, 904784, 1150142, 1457136, 1841200, 2320128

Offset: 1

Author

Michael Somos, Aug 08 2012

Keywords

• nonn

Comments

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

Examples

			q + 4*q^2 + 8*q^3 + 16*q^4 + 30*q^5 + 48*q^6 + 80*q^7 + 128*q^8 + 197*q^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A107035, A215346, A215349.

Programs

Mathematica

nmax=60; CoefficientList[Series[Product[((1+x^k)^3 * (1-x^k) * (1+x^(8*k))^2 / (1-x^(8*k)))^2,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]*EllipticTheta[2, 0, q^4]/(2*EllipticTheta[3, 0, -q]*EllipticTheta[3, 0, q^4]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 07 2017 *)

PARI

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)^3))^2, n))}

Formula

Expansion of q * (f(q) * f(-q^16) / (f(-q) * f(q^4)))^2 = q * (chi(-q^2) * chi(-q^4) / (chi(-q) * chi(-q^8))^2)^2 in powers of q where chi(), f() are Ramanujan theta functions.

Expansion of (eta(q^2)^3 * eta(q^16)^2 / (eta(q)^2 * eta(q^8)^3))^2 in powers of q.

Euler transform of period 16 sequence [ 4, -2, 4, -2, 4, -2, 4, 4, 4, -2, 4, -2, 4, -2, 4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A215346.

a(n) = -(-1)^n * A215349(n). a(2*n) = 4 * A107035(n). Convolution inverse of A215346.

a(n) ~ exp(sqrt(n)*Pi) / (8*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Oct 13 2015