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%I M0600 #28 Apr 03 2018 03:34:34
%S 1,-2,-3,-4,22,30,-12,-128,-147,132,548,516,-552,-1924,-1572,1784,
%T 5790,4410,-5180,-15608,-11406,13712,39128,27528,-33518,-92682,-63156,
%U 77284,208636,139026,-170272,-449904,-294741,360872,936836,604440,-739228,-1892636
%N Expansion of a modular function.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A006709/b006709.txt">Table of n, a(n) for n = -2..1000</a>
%H Morris Newman, <a href="http://plms.oxfordjournals.org/content/s3-9/3/373.extract">Construction and application of a class of modular functions. II</a>. Proc. London Math. Soc. (3) 9 1959 373-387. MR0107629 (21 #6354).
%H Morris Newman, <a href="/A002507/a002507.pdf">Construction and application of a class of modular functions, II</a>, Proc. London Math. Soc. (3) 9 1959 373-387. [Annotated scanned copy, barely legible]
%F Expansion of ((1/q) * f(-q, -q^2)^2 * f(q, q^2) / psi(q^3)^3)^2 in powers of q where psi(), f(,) are Ramanujan theta functions. - _Michael Somos_, Jan 18 2015
%F Expansion of (eta(q) * eta(q^2))^2 * (eta(q^3)^10 / eta(q^6)^14) in powers of q. - _Michael Somos_, Sep 22 2005
%F Euler transform of period 6 sequence [ -2, -4, -12, -4, -2, 0, ...]. - _Michael Somos_, Sep 22 2005
%F G.f.: (x^-2) * (Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k))^2 * (1 - x^(3*k))^10 / (1 - x^(6*k))^14).
%e G.f. = 1/q^2 - 2/q - 3 - 4*q + 22*q^2 + 30*q^3 - 12*q^4 - 128*q^5 +...
%t a[ n_] := SeriesCoefficient[ ( (1/q) QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^3]^5 / QPochhammer[ q^6]^7)^2, {q, 0, n}]; (* _Michael Somos_, Jan 18 2015 *)
%t a[ n_] := SeriesCoefficient[ ( EllipticTheta[ 4, 0, q^3]^2 EllipticTheta[ 4, 0, q^9] (2 / (EllipticTheta[ 4, 0, q^9] - EllipticTheta[ 4, 0, q]))^3)^2, {q, 0, 3 n}]; (* _Michael Somos_, Jan 18 2015 *)
%o (PARI) {a(n) = local(A); if ( n<-2, 0, n+=2; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A)^2 * eta(x^3 + A)^10 / eta(x^6 + A)^14, n))}; /* _Michael Somos_, Sep 22 2005 */
%o (PARI) q='q+O('q^99); Vec((eta(q)*eta(q^2))^2*(eta(q^3)^10/eta(q^6)^14)) \\ _Altug Alkan_, Apr 03 2018
%K sign,easy
%O -2,2
%A _N. J. A. Sloane_
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