OFFSET
0,3
COMMENTS
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of f(x, x^5) / phi(-x^2) in powers of x where phi(), f(, ) are Ramanujan theta functions.
Expansion of q^(-1/3) * eta(q^3) * eta(q^12) / (eta(q) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, ...].
G.f.: Sum_{k>=0} x^(k*(k+1)) * (-x^2, x^2)_k / (x, x)_{2*k+1}.
a(n) ~ Pi * BesselI(1, Pi*sqrt(3*n+1) / sqrt(6)) / (4*sqrt(3*n+1)) ~ exp(sqrt(n/2)*Pi) / (2^(9/4)*sqrt(3)*n^(3/4)) * (1 + (Pi/6 - 3/(4*Pi))/sqrt(2*n) + (Pi^2/144 - 15/(64*Pi^2) - 5/16)/n). - Vaclav Kotesovec, Apr 18 2016, extended Jan 10 2017
EXAMPLE
G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 10*x^7 + 15*x^8 + ...
G.f. = q + q^4 + 2*q^7 + 2*q^10 + 4*q^13 + 5*q^16 + 8*q^19 + 10*q^22 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1-x^(3*k)) * (1+x^(6*k)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 18 2016 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)), n))};
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Michael Somos, Apr 10 2016
STATUS
approved