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A123884
Expansion of phi(x) * phi(-x^6) / chi(-x^2) in powers of x where phi(), chi() are Ramanujan theta functions.
20
1, 2, 1, 2, 3, 2, 2, 0, 2, 2, 1, 4, 0, 2, 3, 2, 2, 0, 4, 2, 2, 0, 0, 2, 1, 4, 2, 2, 2, 2, 3, 2, 0, 2, 2, 2, 2, 0, 2, 4, 4, 0, 0, 0, 1, 2, 4, 0, 2, 4, 2, 2, 1, 6, 0, 2, 2, 0, 0, 2, 4, 2, 0, 2, 2, 0, 4, 0, 4, 2, 1, 2, 0, 2, 4, 0, 0, 2, 2, 4, 3, 4, 0, 2, 2, 2, 2
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/12) * eta(q^2)^4 * eta(q^6)^2 / (eta(q)^2 * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -2, ...].
a(n) = A093829(12*n + 1).
a(n) = (-1)^n * A248886(n). a(2*n) = A131961(n). a(2*n + 1) = 2 * A131963(n). - Michael Somos, Oct 01 2015
EXAMPLE
G.f. = 1 + 2*x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + ...
G.f. = q + 2*q^13 + q^25 + 2*q^37 + 3*q^49 + 2*q^61 + 2*q^73 + 2*q^97 + 2*q^109 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^2 EllipticTheta[ 4, 0, x^6] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Oct 01 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x^2, x^2], {x, 0, n}]; (* Michael Somos, Oct 01 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^2 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^12 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 17 2006
STATUS
approved