A215625 - OEIS

%I #22 Mar 12 2021 22:24:46

%S 1,1,2,-1,1,-1,1,-3,-2,3,3,4,-5,1,-2,0,-7,-4,7,8,11,-11,3,-7,3,-17,

%T -11,17,15,24,-24,7,-14,3,-34,-21,33,34,50,-48,13,-27,8,-68,-42,65,62,

%U 91,-92,24,-51,13,-122,-74,118,115,168,-162,44,-91,27,-221,-136

%N Expansion of q^(2/3) * c(q) / c(q^3) in powers of q where c() is a cubic AGM theta function.

%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 349 Entry 2(viii).

%H G. C. Greubel, <a href="/A215625/b215625.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^(2/3) * eta(q^3)^4 / (eta(q) * eta(q^9)^3) in powers of q.

%F Expansion of f(-q^3)^4 / (f(-q) * f(-q^9)^3) = f(-q^4, -q^5) / f(-q, -q^8) + q * f(-q^2, -q^7) / f(-q^4, -q^5) - q * f(-q, -q^8) / f(-q^2, -q^7) in powers of q where f(), f(,) are Ramanujan theta functions.

%F Euler transform of period 9 sequence [ 1, 1, -3, 1, 1, -3, 1, 1, 0, ...].

%F Given g.f. A(x), then B(q) = A(q^3)/q^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^2 * (u * w - v) - u * w * (u + w).

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

%e G.f. = q^-2 + q + 2*q^4 - q^7 + q^10 - q^13 + q^16 - 3*q^19 - 2*q^22 + 3*q^25 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^4 / (QPochhammer[ x] QPochhammer[ x^9]^3), {x, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^4 / (eta(x + A) * eta(x^9 + A)^3), n))};

%o (PARI) q='q+O('q^99); Vec(eta(q^3)^4/(eta(q)*eta(q^9)^3)) \\ _Altug Alkan_, Mar 30 2018

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

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

%K sign

%O 0,3

%A _Michael Somos_, Aug 17 2012