OFFSET
0,3
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015
Expansion of f(x^2, x^4) / f(-x, -x^5) in powers of x where f() is Ramanujan's general theta function. - Michael Somos, Sep 30 2015
Expansion of q^(1/2) * eta(q^3) * eta(q^4) / (eta(q) * eta(q^12)) in powers of q. - Michael Somos, Sep 30 2015
Euler transform of period 12 sequence [1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, ...]. - Michael Somos, Sep 30 2015
a(n) = number of partitions of n into parts == +-1, +-2, +-5 (mod 12). - Michael Somos, Sep 30 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 30 2015
Convolution inverse of A262771. - Michael Somos, Sep 30 2015
EXAMPLE
G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 8*x^8 + ...
G.f. = 1/q + q^3 + 2*q^7 + 2*q^11 + 3*q^15 + 4*q^19 + 5*q^23 + 7*q^27 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1-x^(3*k)) * (1-x^(4*k)) / ((1-x^k) * (1-x^(12*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] QPochhammer[ x^4] / ( QPochhammer[ x] QPochhammer[ x^12]), {x, 0, n}]; (* Michael Somos, Sep 30 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A) / (eta(x + A) * eta(x^12 + A)), n))}; /* Michael Somos, Sep 30 2015 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 28 2005
STATUS
approved