login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A132978 Expansion of q^(-2/3) * (psi(-q^3) / psi(-q)^3) * (c(q^2) / 3) in powers of q where psi() is a Ramanujan theta function and c() is a cubic AGM theta function. 3
1, 3, 7, 15, 32, 63, 114, 201, 350, 591, 967, 1554, 2468, 3855, 5916, 8970, 13471, 20007, 29384, 42771, 61784, 88530, 125838, 177642, 249230, 347484, 481506, 663549, 909788, 1241127, 1684824, 2276781, 3063657, 4105275, 5478698, 7283709, 9648360, 12735471 (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).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-2/3) * (psi(-q^3) / psi(-q)^3) * (c(q^2) / 3) in powers of q where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.
Expansion of psi(-x^3)^3 * f(-x, x^2) / psi(-x)^4 in powers of x where psi(), f(,) are Ramanujan theta functions.
Expansion of q^(-2/3) * (eta(q^2) * eta(q^6))^2 * eta(q^3) * eta(q^12) / ( eta(q)* eta(q^4) )^3 in powers of q.
Euler transform of period 12 sequence [ 3, 1, 2, 4, 3, -2, 3, 4, 2, 1, 3, 0, ...].
a(n) = A132975(3*n + 2).
Convolution of A132974 and A045833.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
EXAMPLE
G.f. = 1 + 3*x + 7*x^2 + 15*x^3 + 32*x^4 + 63*x^5 + 114*x^6 + 201*x^7 + ...
G.f. = q^2 + 3*q^5 + 7*q^8 + 15*q^11 + 32*q^14 + 63*q^17 + 114*q^20 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2^(1/2) x^(-5/8) EllipticTheta[ 3, 0, x^3] QPochhammer[ x, -x] EllipticTheta[ 2, Pi/4, x^(3/2)]^3 / EllipticTheta[ 2, Pi/4, x^(1/2)]^4, {x, 0, n}] // Simplify;
nmax=60; CoefficientList[Series[Product[(1+x^(3*k))^3 * (1-x^(3*k))^4 * (1+x^(6*k)) / ( (1-x^k)^4 * (1+x^k) * (1+x^(2*k))^3), {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)^2 * eta(x^3 + A) * eta(x^6 + A)^2 * eta(x^12 + A) / ( eta(x + A) * eta(x^4 + A))^3, n))};
CROSSREFS
Sequence in context: A147250 A336701 A066175 * A336976 A368346 A117079
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 07 2007
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)