login
A263548
Expansion of f(x, x) * f(x^2, x^10) in powers of x where f(, ) is Ramanujan's general theta function.
7
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, 6, 2, 2, 2, 0, 0
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)^4 * eta(q^6) * eta(q^24) / (eta(q)^2 * eta(q^8) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ 2, -2, 2, -2, 2, -3, 2, -1, 2, -2, 2, -2, 2, -2, 2, -1, 2, -3, 2, -2, 2, -2, 2, -2, ...].
a(n) = A000377(3*n + 2) = A192013(3*n + 2) = A115660(6*n + 4).
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 + ...
G.f. = q^2 + 2*q^5 + q^8 + 2*q^11 + 2*q^14 + 2*q^20 + 2*q^29 + q^32 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^12] QPochhammer[ -x^10, x^12] QPochhammer[ x^12], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^4] EllipticTheta[ 2, Pi/4, 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^2 + A)^4 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A)^2 * eta(x^8 + A) * eta(x^12 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 20 2015
STATUS
approved