OFFSET
0,2
COMMENTS
LINKS
Robert Israel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of phi(x) * phi(-x^3) * phi(-x^6) / chi(-x^2)^3 in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q^2)^2 * eta(q^3)^2 * eta(q^4) * eta(q^6) / (eta(q)^2 * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [2, 0, 0, -1, 2, -3, 2, -1, 0, 0, 2, -3, ...].
a(n) = A298932(2*n).
EXAMPLE
G.f. = 1 + 2*x + 3*x^2 + 4*x^3 + 4*x^4 + 6*x^5 + 5*x^6 + 6*x^7 + 6*x^8 + ...
G.f. = q + 2*q^5 + 3*q^9 + 4*q^13 + 4*q^17 + 6*q^21 + 5*q^25 + 6*q^29 + ...
MAPLE
N:= 100:
S:= series(JacobiTheta3(0, x)*JacobiTheta4(0, x^3)*JacobiTheta4(0, x^6)*expand(QDifferenceEquations:-QPochhammer(-x^2, x^2, floor(N/2)))^3, x, N+1):
seq(coeff(S, x, j), j=0..N); # Robert Israel, Jan 29 2018
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x^2, x^2]^3, {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^6 + A) / (eta(x + A)^2 * eta(x^12 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jan 29 2018
STATUS
approved