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Expansion of q^(-1) * psi(q) / psi(q^3)^3 in powers of q where psi() is a Ramanujan theta function.
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%I #19 Jul 27 2024 01:26:03

%S 1,1,0,-2,-3,0,4,6,0,-10,-12,0,20,24,0,-36,-45,0,64,78,0,-112,-132,0,

%T 189,222,0,-308,-363,0,492,576,0,-778,-900,0,1210,1392,0,-1844,-2121,

%U 0,2776,3180,0,-4144,-4716,0,6114,6936,0,-8914,-10098,0,12884,14550

%N Expansion of q^(-1) * psi(q) / psi(q^3)^3 in powers of q where psi() 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="/A258093/b258093.txt">Table of n, a(n) for n = -1..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 eta(q^2)^2 * eta(q^3)^3 / (eta(q) * eta(q^6)^6) in powers of q.

%F Euler transform of period 6 sequence [1, -1, -2, -1, 1, 2, ...].

%F G.f.: x^-1 * Product_{k>0} ((1 - x^(2*k))^2 * (1 - x^(3*k))^3) / ((1 - x^k) * (1 - x^(6*k))^6).

%F a(3*n + 1) = 0. a(3*n) = A132979(n). a(3*n - 1) = A258092(n).

%F Convolution inverse is A093829. Convolution with A004016 is A258094.

%e G.f. = 1/q + 1 - 2*q^2 - 3*q^3 + 4*q^5 + 6*q^6 - 10*q^8 - 12*q^9 + ...

%t a[ n_] := SeriesCoefficient[ (1/q) QPochhammer[ q^2]^2 QPochhammer[ q^3]^3 / (QPochhammer[ q] QPochhammer[ q^6]^6), {q, 0, n}]; (* _Michael Somos_, May 25 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^6), n))};

%Y Cf. A093829, A132979, A258092, A258094.

%K sign

%O -1,4

%A _Michael Somos_, May 19 2015