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A258587
Expansion of f(-x, -x) * f(x^2, x^10) in powers of x where f(, ) is Ramanujan's general theta function.
1
1, -2, 1, -2, 2, 0, 2, 0, 0, -2, 1, -4, 0, 0, 2, 0, 3, -2, 2, -2, 2, 0, 0, 0, 0, -4, 2, -2, 0, 0, 0, 0, 3, -2, 0, -2, 4, 0, 2, 0, 0, -4, 1, -2, 0, 0, 4, 0, 2, -2, 0, -4, 2, 0, 0, 0, 0, -2, 2, -2, 0, 0, 0, 0, 2, -2, 3, -4, 2, 0, 2, 0, 0, 0, 2, -2, 0, 0, 2, 0, 3
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 phi(-x) * chi(x^2) * psi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-2/3) * eta(q)^2 * eta(q^4)^2 * eta(q^6) * eta(q^24) / (eta(q^2)^2 * eta(q^8) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ -2, 0, -2, -2, -2, -1, -2, -1, -2, 0, -2, -2, -2, 0, -2, -1, -2, -1, -2, -2, -2, 0, -2, -2, ...].
a(n) = (-1)^n * A263548(n) = A128581(3*n + 2) = A190611(3*n + 2).
a(2*n) = A263571(n). a(2*n + 1) = -2 * A128582(n).
EXAMPLE
G.f. = 1 - 2*x + x^2 - 2*x^3 + 2*x^4 + 2*x^6 - 2*x^9 + x^10 - 4*x^11 + 2*x^14 + ...
G.f. = q^2 - 2*q^5 + q^8 - 2*q^11 + 2*q^14 + 2*q^20 - 2*q^29 + q^32 - 4*q^35 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x^2, x^12] QPochhammer[ -x^10, x^12] QPochhammer[ x^12], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x^2, x^4] EllipticTheta[ 2, 0, x^3] / (2^(1/2) x^(3/4)), {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^24 + A) / (eta(x^2 + A)^2 * eta(x^8 + A) * eta(x^12 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Nov 06 2015
STATUS
approved