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 A227696 Expansion of f(x^3)^3 / f(x) in powers of x where f() is a Ramanujan theta function. 7

%I

%S 1,-1,2,0,2,-1,2,0,1,-2,2,0,2,0,2,0,3,-2,0,0,2,-1,2,0,2,-2,2,0,0,0,4,

%T 0,2,-1,2,0,2,-2,0,0,1,-2,2,0,4,0,2,0,0,-2,2,0,2,0,2,0,3,-2,2,0,2,0,0,

%U 0,2,-3,2,0,0,-2,2,0,4,0,2,0,2,0,0,0,2,-2

%N Expansion of f(x^3)^3 / f(x) 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="/A227696/b227696.txt">Table of n, a(n) for n = 0..1000</a>

%H M. 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^(-1/3) * eta(q) * eta(q^4) * eta(q^6)^9 / (eta(q^2) * eta(q^3) * eta(q^12))^3 in powers of q.

%F Euler transform of period 12 sequence [ -1, 2, 2, 1, -1, -4, -1, 1, 2, 2, -1, -2, ...].

%F Moebius transform is period 36 sequence [ 1, -1, -1, -1, -1, 1, 1, 1, 0, 1, -1, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 1, -1, 0, -1, -1, -1, 1, 1, 1, 1, -1, 0, ...].

%F a(n) = b(3*n+1) where b(n) is multiplicative with b(2^e) = - (1 + (-1)^e) / 2 if e>0, b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^3) / 2 if p == 5 (mod 6).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (4/3)^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A226535.

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

%F a(n) = (-1)^n * A033687(n). a(4*n + 3) = 0.

%F a(2*n) = A097195(n). a(4*n) = A123884(n). a(4*n + 1) = - A033687(n). a(4*n + 2) = 2 * A121361(n).

%e G.f. = 1 - x + 2*x^2 + 2*x^4 - x^5 + 2*x^6 + x^8 - 2*x^9 + 2*x^10 + ...

%e G.f. = q - q^4 + 2*q^7 + 2*q^13 - q^16 + 2*q^19 + q^25 - 2*q^28 + 2*q^31 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3]^3 / QPochhammer[ -q], {q, 0, n}]

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^9 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^3, n))}

%Y Cf. A033687, A097195, A121361, A123884, A226535.

%K sign

%O 0,3

%A _Michael Somos_, Sep 22 2013

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Last modified August 6 15:50 EDT 2020. Contains 336255 sequences. (Running on oeis4.)