OFFSET
0,2
COMMENTS
In volume 2 of Raamunjuan's Notebooks is an obscure equation involving t(1-t) on the left and GG' on the right and they both are equal to the g.f. of 1/3 of this sequence. Here t^(1/3) = c(x)/a(x), (1-t)^(1/3) = b(x)/a(x) since a(x)^3 = b(x)^3 + c(x)^3. N.B. The left side was (t(1-t))^(1/3) but the exponent should be (-1/3) instead which is why the equation was so obscure. - Michael Somos, Mar 13 2019
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 179.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 392.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
G. Manco, How to calculate moduli alpha_3n of the Ramanujan's q_3 theory, Mathematics StackExchange, Jan 2017.
FORMULA
Expansion of (27 * x * (b(x)^3 + c(x)^3)^2 / (b(x) * c(x))^3)^(1/3) in powers of x where b(), c() are cubic AGM theta functions, Michael Somos, Jun 16 2012
Convolution cube is A030197.
a(n) ~ exp(4*Pi*sqrt(n)/3) / (sqrt(6)*n^(3/4)). - Vaclav Kotesovec, Nov 07 2015
EXAMPLE
G.f. = 1 + 14*x + 65*x^2 + 156*x^3 + 456*x^4 + 1066*x^5 + 2250*x^6 + 4720*x^7 + ...
T9a = 1/q + 14*q^2 + 65*q^5 + 156*q^8 + 456*q^11 + 1066*q^14 + 2250*q^17 + ...
MATHEMATICA
a[0] = 1; a[n_] := Module[{A = x*O[x]^n}, A = (QPochhammer[x^3 + A] / QPochhammer[x + A])^12; SeriesCoefficient[((1 + 27*x*A)^2/A)^(1/3), n]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 06 2015, adapted from Michael Somos's PARI script *)
CoefficientList[Series[(QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3)^2 / (QPochhammer[x, x]^2*QPochhammer[x^3, x^3]^4), {x, 0, 50}], x] (* Vaclav Kotesovec, Nov 07 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( ((1 + 27 * x * A)^2 / A)^(1/3), n))}; /* Michael Somos, Jun 16 2012 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 27 2000
STATUS
approved