|
|
A259827
|
|
Expansion of phi(x) * f(-x^12)^3 / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.
|
|
2
|
|
|
1, 2, 0, 0, 3, 2, 0, 0, 4, 6, 0, 0, 4, 2, 0, 0, 4, 8, 0, 0, 7, 2, 0, 0, 8, 10, 0, 0, 4, 4, 0, 0, 5, 10, 0, 0, 8, 4, 0, 0, 12, 10, 0, 0, 8, 6, 0, 0, 4, 14, 0, 0, 12, 2, 0, 0, 8, 14, 0, 0, 8, 4, 0, 0, 9, 18, 0, 0, 12, 6, 0, 0, 16, 14, 0, 0, 4, 4, 0, 0, 12, 12
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
Expansion of phi(x) * c(x^4) / (3 * x^(4/3)) in powers of x where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.
Expansion of q^(-4/3) * eta(q^2)^5 * eta(q^12)^3 / (eta(q)^2 * eta(q^4)^3) in powers of q.
Euler transform of period 12 sequence [ 2, -3, 2, 0, 2, -3, 2, 0, 2, -3, 2, -3, ...].
a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = 2 * A259655(n). 6 * a(n) = A259825(3*n + 4).
|
|
EXAMPLE
|
G.f. = 1 + 2*x + 3*x^4 + 2*x^5 + 4*x^8 + 6*x^9 + 4*x^12 + 2*x^13 + ...
G.f. = q^4 + 2*q^7 + 3*q^16 + 2*q^19 + 4*q^28 + 6*q^31 + 4*q^40 + ...
|
|
MATHEMATICA
|
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ x^12]^3 / QPochhammer[ x^4], {x, 0, n}];
|
|
PROG
|
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^12 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)^3), n))};
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|