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A145977 Expansion of c(q^3) / (c(q^3) + c(q^6)) where c() is a cubic AGM function. 2
1, -1, 1, -1, 2, -3, 4, -5, 7, -10, 12, -15, 20, -26, 32, -39, 50, -63, 76, -92, 114, -140, 168, -201, 244, -295, 350, -415, 496, -591, 696, -818, 967, -1140, 1332, -1554, 1820, -2126, 2468, -2861, 3324, -3855, 4448, -5126, 5916, -6816, 7824, -8970, 10292, -11793, 13471, -15372, 17548, -20007 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..500

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of 1 - q * psi(q^9) / psi(q) = phi(-q^9) / (psi(q) * chi(-q^3)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

Expansion of eta(q) * eta(q^6) * eta(q^9)^2 / (eta(q^2)^2 * eta(q^3) * eta(q^18)), in powers of q.

Euler transform of period 18 sequence [ -1, 1, 0, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, 0, 1, -1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (2/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A139032.

G.f.: Product_{k>0} (P(3, x^k) * P(9, x^k)) / (P(4, x^k)^2 * P(18, x^k)) where P(n, x) is the n-th cyclotomic polynomial.

Convolution inverse of A139032.

a(n) = - A124243(n)  unless n=0. a(2*n) = A128129(n) = a(2*n) unless n=0.

a(2*n + 1) = - A132302(n). a(3*n) = A128641(n).

EXAMPLE

G.f. = 1 - q + q^2 - q^3 + 2*q^4 - 3*q^5 + 4*q^6 - 5*q^7 + 7*q^8 - 10*q^9 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 1 - EllipticTheta[ 2, 0, x^(9/2)] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^9 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)), n))};

CROSSREFS

Cf. A124243, A128129, A128641, A132302, A139032.

Sequence in context: A124243 A132975 A213267 * A050729 A117536 A104665

Adjacent sequences:  A145974 A145975 A145976 * A145978 A145979 A145980

KEYWORD

sign

AUTHOR

Michael Somos, Oct 26 2008

STATUS

approved

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Last modified February 20 00:28 EST 2019. Contains 320329 sequences. (Running on oeis4.)