OFFSET
0,10
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 psi(-x) * psi(x^3) / chi(-x^12)^2 = psi(-x) * chi(x^3) * psi(x^12) in powers of x where psi(), chi() are Ramanujan theta functions.
Expansion of q^(-3/2) * eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^24)^2 / (eta(q^2) * eta(q^3) * eta(q^12)^2) in powers of q.
Euler transform of period 24 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, -1, -1, -1, -1, 0, 0, -1, -2, ...].
2 * a(n) = A263577(2*n + 3).
EXAMPLE
G.f. = 1 - x - x^4 + 2*x^9 + x^12 - x^13 - 2*x^16 + 2*x^21 - x^25 - x^28 + ...
G.f. = q^3 - q^5 - q^11 + 2*q^21 + q^27 - q^29 - 2*q^35 + 2*q^45 - q^53 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-13/8) EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, Pi/4, x^3]^2 / QPochhammer[ x^3], {x, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-13/8) EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, 0, x^6] QPochhammer[ -x^3, x^6], {x, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/2) EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)] QPochhammer[ -x^12, x^12]^2, {x, 0, n}];
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)^2 * eta(x^24 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)^2), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Nov 06 2015
STATUS
approved