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A212484
Expansion of c(q^2) * b(q^6) / (b(q) * c(q) * b(q^3) * c(q^3))^(1/2) in powers of q where b(), c() are cubic AGM theta functions.
4
1, 1, 3, 6, 11, 18, 30, 48, 75, 114, 170, 252, 366, 526, 744, 1044, 1451, 1998, 2730, 3700, 4986, 6672, 8876, 11736, 15438, 20207, 26322, 34134, 44072, 56682, 72612, 92680, 117867, 149400, 188758, 237744, 298554, 373838, 466836, 581412, 722266, 895014
OFFSET
0,3
COMMENTS
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500 (terms 0..42 from Michael Somos)
FORMULA
Expansion of eta(q^6)^6 / (eta(q) * eta(q^2) * eta(q^3)^2 * eta(q^9) * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [1, 2, 3, 2, 1, -2, 1, 2, 4, 2, 1, -2, 1, 2, 3, 2, 1, 0, ...].
a(n) = A123629(n) unless n=0.
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(11/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
EXAMPLE
G.f. = 1 + q + 3*q^2 + 6*q^3 + 11*q^4 + 18*q^5 + 30*q^6 + 48*q^7 + 75*q^8 + ...
MATHEMATICA
nmax=60; CoefficientList[Series[Product[(1-x^(6*k))^6 / ((1-x^k) * (1-x^(2*k)) * (1-x^(3*k))^2 * (1-x^(9*k)) * (1-x^(18*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3]^2 QPochhammer[ q^12]^2 / (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^9] QPochhammer[ q^18]), {q, 0, n}]; (* Michael Somos, Oct 24 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^6 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^9 + A) * eta(x^18 + A)), n))};
CROSSREFS
Cf. A123629.
Sequence in context: A152074 A376709 A123629 * A279100 A347415 A053992
KEYWORD
nonn
AUTHOR
Michael Somos, Jun 02 2012
STATUS
approved