0,2

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.

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 475 Entry 7(i).

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)

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.

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))).

Euler transform of period 9 sequence [3,3,-7,3,3,-7,3,3,-4,...].

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.

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.

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.

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)

(PARI) a(n)=if(n<1, n==0, 3*sigma(n)-if(n%9==0, 27*sigma(n/9)))

(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))

(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)

Sequence in context: A108860 A006499 A140979 this_sequence A155504 A022379 A081601

Adjacent sequences: A096723 A096724 A096725 this_sequence A096727 A096728 A096729

nonn

Michael Somos, Jul 06 2004