0,2

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 313, Equ. (14.2.13).

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

G. E. Andrews and B. C. Berndt, Your Hit Parade: The Top Ten Most Fascinating Formulas in Ramanujan's Lost Notebook, Notices Amer. Math. Soc., 55 (No. 1, Jan. 2008), 18-30. See p. 23, Equation (27).

G.f.: Product_{k>0} (1-x^k)^9/(1-x^3)^3 = 1-9*Sum_{k>0} x^k(1-x^k-6x^(2k)-x^(3k)+x^(4k))/(1+x^k+x^(2k))^3.

Expansion of b(q)^3 in powers of q where b() is a cubic AGM analog function.

Euler transform of period 3 sequence [ -9, -9, -6, ...].

G.f. A(x) satisfies 0= f(A(x), A(x^2), A(x^4)) where f(u, v, w)= v^3 +u*w* (u +6*v -8*w).

Expansion of b(q)^3 in powers of q where b() is a cubic AGM function.

Given A = A0 + A1 + A2 is the 3-section, then 0 = A1^3 + A2^3 - 3*A0*A1*A2. A0 = A(q^3) = b(q^3)^3, A1 = -3 * a(q^3)^2 * c(q^3), A2 = 3 * a(q^3) * c(q^3)^2 where a(), b(), c() are cubic AGM functions.

1 - 9*q + 27*q^2 - 9*q^3 - 117*q^4 + 216*q^5 + 27*q^6 - 450*q^7 + ...

(PARI) a(n)=if(n<1, n==0, -9*sumdiv(n, d, d^2*kronecker(-3, d)))

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^9/eta(x^3+A)^3, n))}

-9 * A103440(n) = a(n) unless n=0. 216 * A134340(n) = a(6*n+5).

Sequence in context: A074954 A103952 A103955 this_sequence A010817 A122985 A115148

Adjacent sequences: A109038 A109039 A109040 this_sequence A109042 A109043 A109044

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Michael Somos, Jun 17 2005