OFFSET
1,2
COMMENTS
Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
Expansion of q/(chi(-q)*chi(-q^3))^6 in powers of q where chi() is a Ramanujan theta function.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Euler transform of period 6 sequence [ 6, 0, 12, 0, 6, 0, ...].
G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)= u^2 -v*(1+12*u+64*u*v)
G.f.: x*(Product_{k>0} (1+x^k)*(1+x^(3k)))^6.
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (64 * 2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
Convolution inverse of A121666. - Seiichi Manyama, Mar 30 2017
MATHEMATICA
nmax = 40; Rest[CoefficientList[Series[x * Product[((1+x^k) * (1+x^(3*k)))^6, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)
PROG
(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^6+A)/eta(x+A)/eta(x^3+A))^6, n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 04 2006
STATUS
approved