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A141094 Expansion of b(q) / b(q^2) in powers of q where b() is a cubic AGM function. 1
1, -3, 3, -3, 6, -9, 12, -15, 21, -30, 36, -45, 60, -78, 96, -117, 150, -189, 228, -276, 342, -420, 504, -603, 732, -885, 1050, -1245, 1488, -1773, 2088, -2454, 2901, -3420, 3996, -4662, 5460, -6378, 7404, -8583, 9972, -11565, 13344, -15378, 17748, -20448, 23472, -26910 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

LINKS

Table of n, a(n) for n=0..47.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of chi(-q)^3 / chi(-q^3) in powers of q where chi() is a Ramanujan theta function.

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

Euler transform of period 6 sequence [ -3, 0, -2, 0, -3, 0, ...].

G.f.: Product_{k>0} (1 - x^(2*k-1))^3 / (1 - x^(6*k-3)).

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 - u * (2 - u*v).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (u^2 - 2*u + 4) - v^3 * (u^2 + u + 1).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6) ) where f(u1, u2, u3, u6) = u1 * (u6^2 - u2 * u3) - u6 * (u3^2 - u6*u2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 2 g(t) where q = exp(2 pi i t) and g() is g.f. for A092848.

Empirical : sum(exp(-Pi)^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = (-2+2*3^(1/2))^(1/3). Simon Plouffe, Feb. 20, 2011.

EXAMPLE

1 - 3*q + 3*q^2 - 3*q^3 + 6*q^4 - 9*q^5 + 12*q^6 - 15*q^7 + 21*q^8 + ...

PROG

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

CROSSREFS

-3 * A124243(n) = a(n) unless n=0. -(-1)^n * A132972(n) = a(n). A128128(n) = a(2*n). -3 * A132302(n) = a(2*n + 1).

Convolution inverse of A128128.

Sequence in context: A153004 A214361 A124449 * A132972 A113920 A081848

Adjacent sequences:  A141091 A141092 A141093 * A141095 A141096 A141097

KEYWORD

sign

AUTHOR

Michael Somos, Jun 04 2008, Aug 12 2009

STATUS

approved

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Last modified April 18 03:01 EDT 2014. Contains 240688 sequences.