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%I #15 Mar 12 2021 22:24:43
%S 1,3,1,-2,2,1,-4,-1,-2,0,2,-4,-1,-2,-2,1,0,2,-2,2,0,-4,1,0,2,2,5,0,-2,
%T 0,0,4,-2,0,0,3,4,0,0,2,1,-4,2,-2,0,0,0,2,-2,0,2,3,-2,0,-2,-2,-4,-1,0,
%U 0,0,-4,2,0,4,0,-4,-2,0,-2,-1,0,0,-2,-2,2,-6,1,2,0,0,4,0,-2,2,0,0,-2,-2,-2,2,0,1,0,0,-2,4,0,0,2,1,6,0,2,0
%N Expansion of f(x, x) * f(x, -x^2) 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).
%D J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339, see page 333.
%D H. Kahl, G. Koehler, Components of Hecke theta series, J. Math. Anal. Appl. 232 (1999), no. 2, 312-331, see page 320. MR1683136 (2000e:11051)
%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 f(sqrt(-x)) * f(-sqrt(-x)) in powers of x where f() is a Ramanujan theta function. - _Michael Somos_, Aug 23 2010
%F Expansion of f(x) * phi(x) in powers of x where f() and phi() are Ramanujan theta functions. - _Michael Somos_, Aug 23 2010
%F Expansion of q^(-1/24) * eta(q^2)^8 / (eta(q)^3 * eta(q^4)^3) in powers of q.
%F Euler transform of period 4 sequence [ 3, -5, 3, -2, ...].
%F G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^k)^3 / (1 + x^(2*k))^3.
%F a(n) = (-1)^n * A115110(n).
%e 1 + 3*x + x^2 - 2*x^3 + 2*x^4 + x^5 - 4*x^6 - x^7 - 2*x^8 + 2*x^10 + ...
%e q + 3*q^25 + q^49 - 2*q^73 + 2*q^97 + q^121 - 4*q^145 - q^169 - 2*q^193 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ I x] QPochhammer[ -I x], {x, 0, 2 n}] (* _Michael Somos_, Jul 12 2012 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x] EllipticTheta[ 3, 0, x], {x, 0, n}] (* _Michael Somos_, Jul 12 2012 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 QPochhammer[ -x, x]^3 / QPochhammer[ -x^2, x^2]^3, {x, 0, n}] (* _Michael Somos_, Jul 12 2012 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 / (eta(x + A)^3 * eta(x^4 + A)^3), n))}
%Y Cf. A115110.
%K sign
%O 0,2
%A _Michael Somos_, May 09 2005
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