OFFSET
0,2
COMMENTS
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-3/4) * (eta(q^2)^3 * eta(q^4)^2 / eta(q)^5)^2 in powers of q.
Euler transform of period 4 sequence [10, 4, 10, 0, ...].
G.f.: Product_{n>=0} (1 - x^(2*n + 1))^-10 * (1 - x^(4*n + 2))^-4.
a(n) ~ exp(2*Pi*sqrt(n)) / (256*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019
EXAMPLE
G.f. = 1 + 10*x + 59*x^2 + 270*x^3 + 1045*x^4 + 3582*x^5 + 11194*x^6 + ...
G.f. = q^3 + 10*q^7 + 59*q^11 + 270*q^15 + 1045*q^19 + 3582*q^23 + 11194*q^27 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 QPochhammer[ x^4]^2 / (QPochhammer[ x]^5))^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ x^(-3/4) (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0, x]^2 / 4)^2, {x, 0, n}];
nmax = 20; CoefficientList[Series[Product[(1 + x^k)^10/(1 - x^(4*k - 2))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 31 2019 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 * eta(x^4 + A)^2 / eta(x + A)^5)^2, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 20 2019
STATUS
approved