OFFSET
0,2
COMMENTS
Expansion of a q-series used in construction of j(tau) to j(2tau) iteration.
REFERENCES
H. Cohn, Introduction to the construction of class fields, Cambridge 1985, p. 191
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u+3)^2 - 8*(u+1)*v^2.
a(n) = A014969(2*n) = A139820(2*n) = A189925(4*n) = A212318(4*n) = A232358(4*n). - Michael Somos, Dec 15 2016
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 1/8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007248. - Michael Somos, Dec 15 2016
a(n) ~ exp(2*Pi*sqrt(n))/(16*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017
EXAMPLE
G.f. = 1 + 32*x + 256*x^2 + 1408*x^3 + 6144*x^4 + 22976*x^5 + 76800*x^6 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1 + 32 x (QPochhammer[ x^4] / QPochhammer[ x])^8, {x, 0, n}]; (* Michael Somos, Dec 15 2016 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x^n * O(x); polcoeff( 1 + 32 * x * (eta(x^4 + A) / eta(x + A))^8, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 02 2004
STATUS
approved