A261444 - OEIS

%I #8 Mar 12 2021 22:24:48

%S 1,0,1,2,2,2,0,4,2,0,1,4,4,2,2,4,5,0,2,2,6,4,2,4,6,0,0,6,4,2,4,8,7,0,

%T 2,10,4,6,0,4,6,0,1,6,8,6,4,8,4,0,4,8,10,4,2,8,8,0,2,6,12,4,4,8,8,0,5,

%U 8,6,4,0,8,14,0,2,10,8,10,2,8,11,0,6,6,6

%N Expansion of f(x^3)^2 * f(-x^6)^2 / f(-x^2) in powers of x where f() 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="/A261444/b261444.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^6)^8 / (eta(q^2) * eta(q^3)^2 * eta(q^12)^2) in powers of q.

%F Euler transform of period 12 sequence [ 0, 1, 2, 1, 0, -5, 0, 1, 2, 1, 0, -3, ...].

%F a(n) = A261426(2*n + 1).

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

%e G.f. = q^2 + q^8 + 2*q^11 + 2*q^14 + 2*q^17 + 4*q^23 + 2*q^26 + ...

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

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^8 / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};

%Y Cf. A261426.

%K nonn

%O 0,4

%A _Michael Somos_, Aug 18 2015