A137569 - OEIS

%I #17 Mar 12 2021 22:24:45

%S 1,-1,-1,1,-1,0,2,-1,-1,3,-2,-1,4,-3,-2,5,-4,-2,8,-6,-4,10,-7,-4,14,

%T -10,-6,18,-13,-7,24,-17,-10,30,-21,-12,40,-28,-17,49,-35,-19,63,-44,

%U -26,78,-55,-31,98,-69,-40,120,-84,-47,150,-105,-61,182,-127,-71

%N Expansion of f(-x) / f(-x^3) 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 Seiichi Manyama, <a href="/A137569/b137569.txt">Table of n, a(n) for n = 0..10000</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^(1/12) eta(q) / eta(q^3) in powers of q.

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

%F Given g.f. A(x) then B(q) = A(q^6)^2 / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 4*v^2 + (u^2 - v) * (w^2 + v).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (432 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A000726.

%F G.f.: Product_{k>0} (1 - x^(3*k-1)) * (1 - x^(3*k-2)).

%F a(3*n) = A035943(n). a(3*n + 1) = - A035941(n). a(3*n + 2) = - A035940(n).

%F Convolution inverse of A000726.

%F Convolution square is A112157. Convolution 4th power is A058095. - _Michael Somos_, Oct 08 2015

%F a(2*n) = A263050(n). a(2*n + 1) = - A263051(n).  - _Michael Somos_, Oct 08 2015

%F G.f.: (Product_{k>0} (1 + x^k + x^(2*k)))^-1. - _Michael Somos_, Oct 08 2015

%F a(n) = -(1/n)*Sum_{k=1..n} A046913(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017

%e G.f. = 1 - x - x^2 + x^3 - x^4 + 2*x^6 - x^7 - x^8 + 3*x^9 - 2*x^10 - x^11 + ...

%e G.f. = 1/q - q^11 - q^23 + q^35 - q^47 + 2*q^71 - q^83 - q^95 + 3*q^107 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] / QPochhammer[ x^3], {x, 0, n}]; (* _Michael Somos_, Oct 08 2015 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^3] QPochhammer[ x^2, x^3], {x, 0, n}]; (* _Michael Somos_, Oct 08 2015 *)

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

%Y Cf. A000726, A035940, A035941, A035943, A058095, A112157, A263050, A263051.

%K sign

%O 0,7

%A _Michael Somos_, Jan 26 2008