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%I
%S 1,3,9,12,21,18,36,24,45,12,54,36,84,42,72,72,93,54,36,60,126,96,108,
%T 72,180,93,126,12,168,90,216,96,189,144,162,144,84,114,180,168,270,
%U 126,288,132,252,72,216,144,372,171,279,216,294,162,36,216,360,240,270,180,504
%N Expansion of eta(q^3)^10/(eta(q)eta(q^9))^3 in powers of q.
%D B. C. Berndt, S. H. Chan, Z.-G. Liu and H. Yesilyurt, A new identity for (q;q)10 [inf] with an application to Ramanujan's partition congruence modulo 11, Quart. J. of Math., 55 (2004), pp. 13-30.
%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 475 Entry 7(i).
%D J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012)
%H B. C. Berndt, S. H. Chan, Z.-G. Liu and H. Yesilyurt, <a href="http://www.math.uiuc.edu/~berndt/articles/partitions1.pdf">A new identity for (q;q)10 [inf] with an application to Ramanujan's partition congruence modulo 11</a>.
%F G.f. Prod_{k>0} (1-x^(3k))^10/((1-x^k)(1-x^(9k)))^3 = 1 + Sum_{k>0} k(3x^k/(1-x^k) -27x^(9k)/(1-x^(9k))).
%F Euler transform of period 9 sequence [3,3,-7,3,3,-7,3,3,-4,...].
%F a(n)=3*b(n) where b(n) is multiplicative and b(3^e) = 1+3*(e>0), b(p^e) = (p^(e+1)-1)/(p-1) otherwise.
%F G.f. A(x) satisfies 0=f(A(x),A(x^2),A(x^4)) where f(u,v,w)=u^2*w +4*u*w^2 +v^3 -6*u*v*w.
%F G.f.: b(q^3)^3/b(q) = c(q)^3/(9c(q^3)) = (a(q)^2+3*a(q^3)^2)/4 = (a(q)^2+a(q)b(q)+b(q)^2)/3 where a(q),b(q),c(q) are the three functions in a cubic AGM analogue described by Borwein.
%t CoefficientList[ Series[1 + Sum[k(3x^k/(1 - x^k) - 27x^(9k)/(1 - x^(9k))), {k, 1, 60}], {x, 0, 60}], x] (from Robert G. Wilson v Jul 14 2004)
%o (PARI) a(n)=if(n<1,n==0,3*sigma(n)-if(n%9==0,27*sigma(n/9)))
%o (PARI) a(n)=local(A); if(n<0,0,A=x*O(x^n); polcoeff(eta(x^3+A)^10/(eta(x+A)*eta(x^9+A))^3,n))
%o (PARI) a(n)=polcoeff(sum(k=1,n,k*3*(x^k/(1-x^k)-9*x^(9*k)/(1-x^(9*k))),1+x*O(x^n)),n)
%K nonn
%O 0,2
%A Michael Somos, Jul 06 2004
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