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A128771 Expansion of phi(-q) / phi(-q^9) in powers of q where phi() is a Ramanujan theta function. 3
1, -2, 0, 0, 2, 0, 0, 0, 0, 0, -4, 0, 0, 4, 0, 0, 2, 0, 0, -8, 0, 0, 8, 0, 0, 2, 0, 0, -16, 0, 0, 16, 0, 0, 4, 0, 0, -28, 0, 0, 28, 0, 0, 8, 0, 0, -48, 0, 0, 46, 0, 0, 12, 0, 0, -80, 0, 0, 76, 0, 0, 20, 0, 0, -126, 0, 0, 120, 0, 0, 32, 0, 0, -196, 0, 0, 184, 0, 0, 48, 0, 0, -300, 0, 0, 280, 0, 0, 72, 0, 0 (list; graph; refs; listen; history; text; internal format)
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 eta(q)^2 * eta(q^18) / (eta(q^2) * eta(q^9)^2) in powers of q.
Euler transform of period 18 sequence [ -2, -1, -2, -1, -2, -1, -2, -1, 0, -1, -2, -1, -2, -1, -2, -1, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1-u) * (u-v^2) - 2*u * (v-1).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u-v)^3 - u* (3-u) * (v-1) * (3 - 2*u + u*v).
G.f.: Product_{k>0} (1 - x^k) * (1 + x^(9*k)) / ( (1 + x^k) * (1 - x^(9*k)) ).
a(3n+2)= a(3n+3)= 0.
Empirical : sum(exp(-Pi/3)^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = sqrt(3). Simon Plouffe, Feb. 20, 2011.
Convolution inverse of A128770. a(3*n + 1) = -2*A092848(n).
EXAMPLE
G.f. = 1 - 2*q + 2*q^4 - 4*q^10 + 4*q^13 + 2*q^16 - 8*q^19 + 8*q^22 + 2*q^25 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^9], {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^18 + A) / (eta(x^2 + A) * eta(x^9 + A)^2), n))};
CROSSREFS
Sequence in context: A193531 A093492 A139380 * A000122 A002448 A033759
KEYWORD
sign
AUTHOR
Michael Somos, Mar 27 2007
STATUS
approved

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Last modified June 16 22:16 EDT 2024. Contains 373432 sequences. (Running on oeis4.)