OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(3/8) * eta(q)^2 * eta(q^6) / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -2, -1, -1, -1, -2, -1, -2, -1, -1, -1, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (768 t)) = 12^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132217.
G.f.: 1 / (Product_{k>0} (1 + x^k) * (1 + x^k + x^(2*k)) * (1 + x^(6*k))).
2 * a(n) = A253243(2*n + 3). a(3*n + 2) = 0.
Convolution inverse of A132218.
EXAMPLE
G.f. = 1 - 2*x + x^3 + x^6 - 2*x^10 + x^12 - 2*x^13 + 2*x^15 + x^18 + x^21 + ...
G.f. = 1/q^3 - 2*q^5 + q^21 + q^45 - 2*q^77 + q^93 - 2*q^101 + 2*q^117 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2^(1/2) x^(3/8) EllipticTheta[ 4, 0, x] / EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}];
a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, -q]*QPochhammer[-q^3, q^6]/ QPochhammer[q^6, q^6], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 19 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^6)/(eta(q^2)*eta(q^3)*eta(q^12))) \\ Altug Alkan, Mar 20 2018
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Nov 09 2015
STATUS
approved