A263398 - OEIS

%I #11 Mar 12 2021 22:24:48

%S 1,-2,-7,14,20,-36,-34,40,50,-30,-71,76,82,-144,-98,112,131,-70,-140,

%T 170,168,-288,-228,232,246,-120,-290,258,310,-468,-280,344,337,-190,

%U -350,394,412,-648,-510,496,462,-252,-583,558,602,-864,-532,584,664,-350

%N Expansion of phi(-x^2)^6 * psi(x^6) / f(x)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A263398/b263398.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="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

%F Euler transform of period 12 sequence [-2, -8, -2, -4, -2, -7, -2, -4, -2, -8, -2, -5, ...].

%F 8 * a(n) = A245643(3*n + 2).

%e G.f. = 1 - 2*x - 7*x^2 + 14*x^3 + 20*x^4 - 36*x^5 - 34*x^6 + 40*x^7 + ...

%e G.f. = q^2 - 2*q^5 - 7*q^8 + 14*q^11 + 20*q^14 - 36*q^17 - 34*q^20 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2]^6 EllipticTheta[ 2, 0, x^3] / (2 x^(3/4) QPochhammer[ -x]^2), {x, 0, n}];

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

%o (PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^2)^6*eta(q^12)^2/(eta(q^4)^4*eta(q^6))) \\ _Altug Alkan_, Jul 31 2018

%Y Cf. A245643.

%K sign

%O 0,2

%A _Michael Somos_, Oct 16 2015