A213618 - OEIS

%I #12 Feb 16 2025 08:33:17

%S 1,0,0,-2,0,0,0,0,-3,0,0,6,2,0,0,0,0,0,0,0,-6,0,0,0,6,0,0,-14,0,0,0,0,

%T -3,0,0,12,12,0,0,0,0,0,0,0,-6,0,0,0,2,0,0,-12,0,0,0,0,-12,0,0,18,0,0,

%U 0,0,0,0,0,0,-12,0,0,0,18,0,0,-14,0,0,0,0

%N Expansion of phi(-q^3) * b(q^8) in powers of q where phi() is a Ramanujan theta function and b() is a cubic AGM theta function.

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

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

%H G. C. Greubel, <a href="/A213618/b213618.txt">Table of n, a(n) for n = 0..2500</a>

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

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

%F Expansion of eta(q^3)^2 * eta(q^8)^3 / (eta(q^6) * eta(q^24)) in powers of q.

%F Euler transform of period 24 sequence [ 0, 0, -2, 0, 0, -1, 0, -3, -2, 0, 0, -1, 0, 0, -2, -3, 0, -1, 0, 0, -2, 0, 0, -3, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 93312^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A213607.

%F a(3*n + 1) = a(4*n + 1) = a(4*n + 2) = a(24*n + 15) = a(24*n + 23) = 0.

%F a(12*n) = A014452(n). a(24*n + 8) = -3 * A213592(n). a(24*n + 11) = 6 * A213617(n). a(24*n + 20) = -6 * A213607(n).

%e G.f. = 1 - 2*q^3 - 3*q^8 + 6*q^11 + 2*q^12 - 6*q^20 + 6*q^24 - 14*q^27 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3] QPochhammer[ q^8]^3 / QPochhammer[ q^24], {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^8 + A)^3 / (eta(x^6 + A) * eta(x^24 + A)), n))};

%Y Cf. A014452, A213607, A213617.

%K sign,changed

%O 0,4

%A _Michael Somos_, Jun 16 2012