A138270 - OEIS

%I #11 Mar 12 2021 22:24:45

%S 1,0,0,-2,-2,0,0,4,0,0,0,0,2,0,0,0,-2,0,0,-4,0,0,0,0,0,0,0,-2,4,0,0,4,

%T 0,0,0,0,-2,0,0,4,0,0,0,-4,0,0,0,0,-2,0,0,0,-4,0,0,0,0,0,0,0,0,0,0,4,

%U 6,0,0,-4,0,0,0,0,0,0,0,-2,4,0,0,4,0,0,0

%N Expansion of phi(-q^3) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

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

%H G. C. Greubel, <a href="/A138270/b138270.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 (eta(q^3) * eta(q^4))^2 / (eta(q^6) * eta(q^8)) in powers of q.

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

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 192^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112609.

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

%F a(4*n) = A164273(n). - _Michael Somos_, Sep 27 2015

%e G.f. = 1 - 2*q^3 - 2*q^4 + 4*q^7 + 2*q^12 - 2*q^16 - 4*q^19 - 2*q^27 + 4*q^28 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3] EllipticTheta[ 4, 0, q^4], {q, 0, n}]; (* _Michael Somos_, Sep 27 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^4 + A))^2 / (eta(x^6 + A) * eta(x^8 + A)), n))};

%Y Cf. A112609, A164273.

%K sign

%O 0,4

%A _Michael Somos_, Mar 10 2008, Apr 04 2008