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%I #25 Mar 12 2021 22:24:43
%S 1,-4,1,16,-24,-4,50,-64,1,96,-120,16,170,-200,-24,256,-288,-4,362,
%T -384,50,480,-528,-64,601,-680,1,800,-840,96,962,-1024,-120,1152,
%U -1200,16,1370,-1448,170,1536,-1680,-200,1850,-1920,-24,2112,-2208,256,2451,-2404,-288,2720,-2808,-4
%N Expansion of eta(q)^4 * eta(q^2) * eta(q^6)^5 / eta(q^3)^4 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 226 Entry 4(i).
%H G. C. Greubel, <a href="/A111661/b111661.txt">Table of n, a(n) for n = 1..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 Euler transform of period 6 sequence [-4, -5, 0, -5, -4, -6, ...].
%F Expansion of q * psi(q)^2 * phi(-q)^3 * psi(q^3)^2 / phi(-q^3) in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Mar 01 2011
%F Expansion of (b(q^2)^3 - b(q)^3) / 9 in powers of q where b() is a cubic AGM theta function. - _Michael Somos_, Mar 01 2011
%F Expansion of (1/3) * b(q) * b(q^2) * c(q^2)^2 / c(q) in powers of q where b(), c() are cubic AGM theta functions. - _Michael Somos_, Jul 09 2012
%F a(n) is multiplicative with a(2^e) = (-4)^e, a(3^e) = 1, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 6), a(p^e) = (1 - (-p^2)^(e+1)) / (p^2 + 1) if p == 5 (mod 6). - _Michael Somos_, Mar 01 2011
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 243^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A214262.
%F G.f.: Sum_{k>0} Kronecker(k, 3) * k^2 * x^k / (1 - x^(2*k)) = x * Product_{k>0} (1 - x^k)^4 * (1 - x^(2*k)) * (1 + x^(3*k))^5 * (1 - x^(3*k)).
%e G.f. = q - 4*q^2 + q^3 + 16*q^4 - 24*q^5 - 4*q^6 + 50*q^7 - 64*q^8 + q^9 + ...
%t a[ n_]:= If[ n < 1, 0, Sum[ Mod[ n/d, 2] d^2 KroneckerSymbol[d, 3], {d, Divisors[n]}]]; (* _Michael Somos_, Jul 09 2012 *)
%t eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[eta[q]^4* eta[q^2]*eta[q^6]^5/eta[q^3]^4, {q, 0, 30}], q] (* _G. C. Greubel_, Apr 18 2018 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^2 * kronecker(d,3)) )};
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A) * eta(x^6 + A)^5 / eta(x^3 + A)^4, n))};
%K sign,mult
%O 1,2
%A _Michael Somos_, Aug 08 2005
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