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A232772 Expansion of (psi(x)^2 / (phi(-x) * phi(x^2)))^2 in powers of x where phi(), psi() are Ramanujan theta functions. 2
1, 8, 30, 80, 197, 472, 1046, 2160, 4306, 8360, 15712, 28656, 51127, 89552, 153926, 259904, 432336, 709728, 1150142, 1841200, 2915546, 4570904, 7097622, 10921184, 16664073, 25228176, 37907758, 56553936, 83806768, 123405752, 180611558, 262799248, 380275604 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..32.

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-1/2) * (eta(q^2)^7 * eta(q^8)^2 / (eta(q)^4 * eta(q^4)^5))^2 in powers of q.

Euler transform of period 8 sequence [ 8, -6, 8, 4, 8, -6, 8, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 g(t) where q = eqp(2 Pi i t) and g() is the g.f. of A233458.

a(n) = A215349(2*n + 1) = A215348(2*n + 1). 2 * a(n) = A212318(2*n + 1) = - A232358(2*n + 1).

a(n) ~ exp(sqrt(2*n)*Pi) / (2^(17/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

EXAMPLE

G.f. = 1 + 8*x + 30*x^2 + 80*x^3 + 197*x^4 + 472*x^5 + 1046*x^6 + 2160*x^7 + ...

G.f. = q + 8*q^3 + 30*q^5 + 80*q^7 + 197*q^9 + 472*q^11 + 1046*q^13 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^4 / (16 q^(1/2)) / (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]

nmax=60; CoefficientList[Series[Product[((1-x^k)^3 * (1+x^k)^7 * (1+x^(4*k))^2 / (1-x^(4*k))^3)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 * eta(x^8 + A)^2 / (eta(x + A)^4 * eta(x^4 + A)^5))^2, n))}

CROSSREFS

Cf. A212318, A215348, A215348, A232358, A233458.

Sequence in context: A299284 A002417 A126858 * A213776 A113751 A107233

Adjacent sequences:  A232769 A232770 A232771 * A232773 A232774 A232775

KEYWORD

nonn

AUTHOR

Michael Somos, Nov 30 2013

STATUS

approved

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Last modified February 22 19:36 EST 2018. Contains 299469 sequences. (Running on oeis4.)