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A233006
Expansion of psi(x) / f(-x^6) in powers of x where psi(), f() are Ramanujan theta functions.
2
1, 1, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 3, 2, 0, 3, 1, 0, 5, 3, 0, 5, 2, 0, 8, 5, 0, 8, 4, 0, 12, 7, 0, 12, 6, 0, 19, 11, 0, 19, 9, 0, 27, 15, 0, 28, 14, 0, 39, 22, 0, 41, 20, 0, 55, 31, 0, 58, 29, 0, 77, 43, 0, 82, 41, 0, 106, 58, 0, 113, 57, 0, 145, 80, 0, 156
OFFSET
0,7
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^(-1/8) * eta(q^2)^2 / (eta(q) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ 1, -1, 1, -1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = (3/2)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A070047.
G.f.: Product_{k>0} (1 + x^k) / (1 + x^(2*k) + x^(4*k)).
a(3*n) = A070047(n). a(3*n + 1) = A097451(n). a(3*n + 2) = 0.
EXAMPLE
G.f. = 1 + x + x^3 + 2*x^6 + x^7 + x^9 + x^10 + 3*x^12 + 2*x^13 + 3*x^15 + ...
G.f. = q + q^9 + q^25 + 2*q^49 + q^57 + q^73 + q^81 + 3*q^97 + 2*q^105 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8) QPochhammer[ x^6]), {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 + A) * eta(x^6 + A)), n))};
CROSSREFS
Sequence in context: A255361 A341685 A069844 * A145152 A124327 A346837
KEYWORD
nonn
AUTHOR
Michael Somos, Dec 03 2013
STATUS
approved