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

1,2

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 = 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

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

Example

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 + ...

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 *)

Prog

(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))}

Crossrefs

Cf. A107035, A215346, A215349.

Sequence in context: A301149 A301143 A215349 * A301144 A298803 A320947

Adjacent sequences: A215345 A215346 A215347 * A215349 A215350 A215351

Keyword

nonn

Author

Michael Somos, Aug 08 2012

Status

approved