A263773 - OEIS

%I #14 Mar 12 2021 22:24:48

%S 1,6,9,-12,-42,-18,36,48,45,-12,-108,-36,84,84,72,-72,-186,-54,36,120,

%T 126,-96,-216,-72,180,186,126,-12,-336,-90,216,192,189,-144,-324,-144,

%U 84,228,180,-168,-540,-126,288,264,252,-72,-432,-144,372,342,279,-216

%N Expansion of b(-q)^2 in powers of q where b() is a cubic AGM theta function.

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

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

%H G. C. Greubel, <a href="/A263773/b263773.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 f(q)^6 / f(q^3)^2 in powers of q where f() is a Ramanujan theta function.

%F Expansion of (eta(q^2)^9 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)^3))^2 in powers of q.

%F Euler transform of period 12 sequence [ 6, -12, 4, -6, 6, -8, 6, -6, 4, -12, 6, -4, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 972 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134079.

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

%F a(2*n + 1) = 6 * A252651(n). a(3*n + 2) = 9 * A134079(n).

%F Convolution square of A226535.

%e G.f. = 1 + 6*x + 9*x^2 - 12*x^3 - 42*x^4 - 18*x^5 + 36*x^6 + 48*x^7 + 45*x^8 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^6 / QPochhammer[ -q^3]^2, {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^9 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^3))^2, n))};

%Y Cf. A134079, A226535, A262651.

%K sign

%O 0,2

%A _Michael Somos_, Oct 27 2015