OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of ( f(x^3)^3 / f(x) )^2 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-2/3) * eta(q)^2 * eta(q^4)^2 * eta(q^6)^18 / (eta(q^2) * eta(q^3)* eta(q^12))^6 in powers of q.
Euler transform of period 12 sequence [ -2, 4, 4, 2, -2, -8, -2, 2, 4, 4, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (4/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263773.
From Michael Somos, Feb 19 2015: (Start)
a(2*n + 1) = -2 * A098098(n).
Convolution square of A227696. (End)
Sum_{k=1..n} a(k) ~ (Pi^2/54) * n^2. - Amiram Eldar, Nov 23 2023
EXAMPLE
G.f. = 1 - 2*x + 5*x^2 - 4*x^3 + 8*x^4 - 6*x^5 + 14*x^6 - 8*x^7 + 14*x^8 - ...
G.f. = q^2 - 2*q^5 + 5*q^8 - 4*q^11 + 8*q^14 - 6*q^17 + 14*q^20 - 8*q^23 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3]^3 / QPochhammer[ -x])^2, {x, 0, n}]; (* Michael Somos, Feb 19 2015 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[ 1, 3 n + 2] / 3]; (* Michael Somos, Feb 19 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( ( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^9 / ( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) )^3 )^2, n))};
(PARI) {a(n) = if( n<0, 0, (-1)^n * sigma(3*n + 2) / 3)}; /* Michael Somos, Feb 19 2015 */
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 06 2007
STATUS
approved