OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q * (psi(q)^3 * phi(-q)^2)^4 * ((phi(q) / psi(q))^4 + 16 * q * (psi(q) / phi(q))^4) in powers of q where phi(), psi() are Ramanujan theta functions.
a(n) is multiplicative with a(2^e) = 16^e, a(p^e) = a(p) * a(p^(e-1)) - p^9 * a(p^(e-2)) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 32 (t / i)^10 f(t) where q = exp(2 Pi i t).
EXAMPLE
G.f. = q + 16*q^2 - 156*q^3 + 256*q^4 + 870*q^5 - 2496*q^6 - 952*q^7 + 4096*q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2]^7 / QPochhammer[ q^4]^2)^4 + 16 q^2 (QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^2)^4, {q, 0, n}]; (* Michael Somos, May 28 2013 *)
PROG
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / eta(x^4 + A)^2)^4 + 16 * x * (eta(x + A)^2 * eta(x^2 + A) * eta(x^4 + A)^2)^4, n))};
(PARI) q='q+O('q^99); Vec((eta(q^2)^7/eta(q^4)^2)^4+16*q*(eta(q)^2*eta(q^2)*eta(q^4)^2)^4) \\ Altug Alkan, Apr 18 2018
(Sage) CuspForms( Gamma1(2), 10, prec=50).0; # Michael Somos, May 28 2013
(Magma) Basis( CuspForms( Gamma1(2), 10), 50) [1]; /* Michael Somos, May 27 2014 */
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Michael Somos, Feb 23 2010
EXTENSIONS
Name corrected by Altug Alkan, Apr 18 2018
STATUS
approved