A230442 - OEIS

%I #24 Mar 12 2021 22:24:47

%S 1,-2,-2,4,1,2,-2,-4,-1,-4,6,0,0,6,4,-4,-4,2,-6,0,-5,2,0,0,4,2,6,4,-1,

%T -6,2,0,4,-6,-8,-8,8,-2,-6,8,-4,4,4,4,4,-2,-2,8,-1,4,-4,0,-4,-8,-6,0,

%U 0,0,6,-8,-3,-2,6,-4,8,12,-2,-4,4,0,10,4,-4,-2,0,-8,-4,-2,4,4,-12,2,-4,0,-12,4,-4

%N Expansion of q^(-1/6) * eta(q)^2 * eta(q^2) in powers of q.

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

%H G. C. Greubel, <a href="/A230442/b230442.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(-x)^3 / chi(-x) = f(-x^2)^3 * chi(-x)^2 = phi(-x) * f(-x^2)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

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

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

%F Convolution square is A230280.

%e G.f. = 1 - 2*x - 2*x^2 + 4*x^3 + x^4 + 2*x^5 - 2*x^6 - 4*x^7 - x^8 + ...

%e G.f. = q - 2*q^7 - 2*q^13 + 4*q^19 + q^25 + 2*q^31 - 2*q^37 - 4*q^43 + ...

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

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

%o (PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^2)) \\ _Altug Alkan_, Apr 18 2018

%Y Cf. A230280.

%K sign

%O 0,2

%A _Michael Somos_, Oct 18 2013

%E Terms and offset corrected by _Joerg Arndt_, Oct 21 2013