A102315 Expansion of (b(q^6) * c(q^6)) / (b(q^3) * c(q^3)) in powers of q where b(), c() are cubic AGM theta functions.
1, 2, 3, 8, 13, 20, 37, 56, 83, 134, 196, 280, 419, 592, 824, 1176, 1618, 2202, 3040, 4096, 5471, 7368, 9753, 12824, 16937, 22090, 28653, 37248, 47968, 61488, 78887, 100472, 127461, 161702, 203951, 256368, 322090, 402748, 502112, 625464, 776061
Offset: 0
Keywords
Examples
G.f. = 1 + 2*x + 3*x^2 + 8*x^3 + 13*x^4 + 20*x^5 + 37*x^6 + 56*x^7 + ... G.f. = q + 2*q^4 + 3*q^7 + 8*q^10 + 13*q^13 + 20*q^16 + 37*q^19 + ...
References
- B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag. See p. 179, equation (13.22).
- S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 392.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] QPochhammer[ x^3, x^6])^-2, {x, 0, n}]; (* Michael Somos, Feb 19 2015 *) nmax = 60; CoefficientList[Series[Product[(1+x^k)^2 * (1+x^(3*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A) / (eta(x + A) * eta(x^3 + A)))^2, n))}; -
PARI
q='q+O('q^99); Vec((eta(q^2)*eta(q^6)/(eta(q)*eta(q^3)))^2) \\ Altug Alkan, Apr 21 2018
Formula
Expansion of (chi(-x) * chi(-x^3))^(-2) in powers of x where chi() is a Ramanujan theta function.
Euler transform of period 6 sequence [2, 0, 4, 0, 2, 0, ...].
Expansion of q^(-1) * (eta(q^2) * eta(q^6) / (eta(q) * eta(q^3)))^2 in powers of q^3.
Convolution inverse of A058543. - Michael Somos, Feb 19 2015
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(11/4)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 08 2015
Comments