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 (1/q) * chi(-q) * chi(q)^3 * chi(-q^5)^3 * chi(q^5) in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q^2)^5 * eta(q^5)^2 / (eta(q)^2 * eta(q^4)^3 * eta(q^10) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [2, -3, 2, 0, 0, -3, 2, 0, 2, -4, 2, 0, 2, -3, 0, 0, 2, -3, 2, 0, ...].
EXAMPLE
G.f. = q^-1 + 2 + 3*q^3 - 4*q^5 + 4*q^7 - 4*q^9 + 7*q^11 - 12*q^13 + 13*q^15 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1/q EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^5] / (QPochhammer[ q^4] QPochhammer[ q^20]), {q, 0, n}];
a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q, q^2] QPochhammer[ -q, q^2]^3 QPochhammer[ q^5, q^10]^3 QPochhammer[ -q^5, q^10], {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^5 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^10 + A) * eta(x^20 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jan 15 2018
STATUS
approved