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
Euler transform of period 6 sequence [ -7, -2, -2, -2, -7, -4, ...].
Expansion of q * (phi(-q) * psi(q^3))^3 / (phi(-q^3) * psi(q)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (b(q)^2 / b(q^2)) * (c(q^2)^2 / c(q)) / 3 in powers of q where b(), c() are cubic AGM theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 6 (t / i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 19 2013
G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 -x^(3*k))^2 * (1 + x^(3*k))^7 / (1 + x^k)^5.
EXAMPLE
G.f. = q - 7*q^2 + 19*q^3 -23*q^4 + 6*q^5 + 11*q^6 + 8*q^7 - 55*q^8 + ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, Sum[ { 1, -4, 6, -4, 1, 0}[[ Mod[ d, 6, 1]]] d, {d, Divisors[ n]}]]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^6])^7 / (QPochhammer[ q^2] QPochhammer[ q^3])^5, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)
PROG
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d*[ 0, 1, -4, 6, -4, 1][ d%6 + 1]))};
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A))^7 / (eta(x^2 + A) * eta(x^3 + A))^5, n))};
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, n/d *[ 0, 1, -9, 16, -9, 1][ d%6 + 1]))}; /* Michael Somos, Sep 19 2013 */
(Sage) A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[1] - 7*A[2]; # Michael Somos, Sep 19 2013
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 02 2006
STATUS
approved