login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A123861 Expansion of (f(q) * f(q^3) / (f(-q) * f(-q^3)))^2 in powers of q where f() is a Ramanujan theta function. 3
1, 4, 8, 20, 48, 88, 168, 320, 544, 932, 1584, 2544, 4080, 6488, 9984, 15288, 23232, 34568, 51144, 75152, 108832, 156736, 224352, 317728, 447648, 627292, 871856, 1206068, 1660416, 2271032, 3092976, 4194464, 5657728, 7602096, 10175760 (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

G. C. Greubel, Table of n, a(n) for n = 0..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.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of phi(q) * phi(q^3) / (phi(-q) * phi(-q^3)) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Aug 31 2014

Expansion of eta(q^2)^6 * eta(q^6)^6 / (eta(q)^4 * eta(q^3)^4 * eta(q^4)^2 * eta(q^12)^2) in powers of q. - Michael Somos, Aug 31 2014

Euler transform of period 12 sequence [4, -2, 8, 0, 4, -4, 4, 0, 8, -2, 4, 0, ...].

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

Let g.f. A(x) = u, then B(x) = u * (u-1) / 4, B(x^2) = ((u-1) / 4)^2 / u where B(x) is the g.f. for A123653.

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187197. - Michael Somos, Aug 31 2014

a(n) = 4 * A123647(n) unless n=0.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

EXAMPLE

G.f. = 1 + 4*q + 8*q^2 + 20*q^3 + 48*q^4 + 88*q^5 + 168*q^6 + 320*q^7 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] QPochhammer[ -q^3] / (QPochhammer[ q] QPochhammer[ q^3]))^2, {q, 0, n}]; (* Michael Somos, Aug 31 2014 *)

nmax=60; CoefficientList[Series[Product[(1+x^k)^6 * (1-x^k)^2 * (1+x^(3*k))^4 / ((1-x^(4*k))^2 * (1+x^(6*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^6 + A)^6 / (eta(x + A)^4 * eta(x^3 + A)^4 * eta(x^4 + A)^2 * eta(x^12 + A)^2), n))};

CROSSREFS

Cf. A123647, A187197.

Sequence in context: A190589 A009916 A203167 * A115099 A060919 A009333

Adjacent sequences:  A123858 A123859 A123860 * A123862 A123863 A123864

KEYWORD

nonn

AUTHOR

Michael Somos, Oct 14 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 23 17:30 EST 2020. Contains 338595 sequences. (Running on oeis4.)