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%I #35 Mar 12 2021 22:24:41
%S 1,8,20,0,-62,0,216,0,-641,0,1636,0,-3778,0,8248,0,-17277,0,34664,0,
%T -66878,0,125312,0,-229252,0,409676,0,-716420,0,1230328,0,-2079227,0,
%U 3460416,0,-5677816,0,9198424,0,-14729608,0,23328520,0,-36567242,0,56774712,0
%N Expansion of 16/lambda(z) in powers of nome q = exp(Pi*i*z).
%C Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).
%H Seiichi Manyama, <a href="/A029845/b029845.txt">Table of n, a(n) for n = -1..10000</a> (terms -1..1000 from Alois P. Heinz)
%H J. H. Conway and S. P. Norton, <a href="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.
%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>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EllipticLambdaFunction.html">Elliptic Lambda Function</a>
%F Expansion of (eta(q^2)^3/(eta(q)*eta(q^4)^2))^8 in powers of q. - _Michael Somos_, Nov 14 2006
%F Expansion of (chi(q)*chi(-q^2))^8/q in powers of q where chi() is a Ramanujan theta function.
%F Euler transform of period 4 sequence [ 8, -16, 8, 0, ...]. - _Michael Somos_, Nov 14 2006
%F G.f. A(x) satisfies: 0=f(A(x), A(x^2)) where f(u, v) = 256 - v*(32-16*u+u^2) + v^2. - _Michael Somos_, Nov 14 2006
%F G.f.: 1/q*(Product_{k>0} (1+q^(2k-1))/(1+q^(2k)))^8.
%e 1/q + 8 + 20*q - 62*q^3 + 216*q^5 - 641*q^7 + 1636*q^9 - 3778*q^11 + ...
%t QP = QPochhammer; s = 16*q + (QP[q]/QP[q^4])^8 + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015, adapted from PARI *)
%o (PARI) {a(n)=local(A); if(n<-1, 0, n++; A=x*O(x^n); polcoeff( 16*x+(eta(x+A)/eta(x^4+A))^8, n))} /* _Michael Somos_, Nov 14 2006 */
%Y Cf. A007248(n) = a(2n-1). A124972(n) = a(n) except at n=0.
%Y Cf. A000122, A000700, A010054, A121373.
%K sign,easy
%O -1,2
%A _N. J. A. Sloane_
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