0,2

Denoted by g_4(q) in Cynk and Hulek in Remark 3.4 on page 12

a(n)=0 if and only if A033687(n)=0 (see the Han-Ono paper). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 16 2008

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)

Newman, Morris; A table of the coefficients of the powers of $\eta(\tau)$. Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.

G.-N. Han and Ken Ono, Hook lengths and 3-cores (available at http://www-irma.u-strasbg.fr/~guoniu/hook/hh3core).

S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).

Index entries for expansions of Product_{k >= 1} (1-x^k)^m

S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds

W. Stein, Modular Forms Database.

Expansion of q^(-1)eta(q^3)^8 in powers of q^3.

Euler transform of period 1 sequence [ -8, ...].

a(4n+1)=-8a(n). - Michael Somos Dec 06 2004

G.f.: Product_{k>0} (1-x^k)^8.

a(4n+3)=a(16n+13)=0. - Michael Somos Oct 19 2005

a(n)=b(3n+1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2*(-1)^(e/2)*p^(3e/2) if p == 2 (mod 3), b(p^e) = b(p)b(p^(e-1)) -b(p^(e-2))p^3 if p == 1 (mod 3) where b(p) = (x^2-3p)x, 4p = x^2+3y^2, |x|<|y|, and x == 2 (mod 3). - Michael Somos Aug 23 2006

Expansion of q^(-1/3)b(q)^3*c(q)/3 in powers of q. - Michael Somos Nov 08 2006

Expansion of q^(-1)b(q)*c(q)^3/27 in powers of q^3. - Michael Somos Nov 08 2006

Given g.f. A(x), then B(x)= x*A(x^3) satisfies 0= f(B(x), B(x^2), B(x^4)) where f(u, v, w)= v^3 -u*w*(u +16*w) . - Michael Somos Feb 19 2007

A092342(n) = a(n) + 81*A033690(n-1). - Michael Somos Aug 22 2007

eta(q^3)^8 = q - 8*q^4 + 20*q^7 - 70*q^13 + 64*q^16 + 56*q^19 ...

q - 8*q^4 + 20*q^7 - 70*q^13 + 64*q^16 + 56*q^19 - 125*q^25 - ...

(PARI) {a(n)=local(A, p, e, x, y, a0, a1); if(n<0, 0, n=3*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3, 0, if(p%3==2, if(e%2, 0, (-1)^(e/2)*p^(3*e/2)), forstep(y=sqrtint(4*p\3), sqrtint(p\3), -1, if(issquare(4*p-3*y^2, &x), if(x%3!=2, x=-x); break)); a0=1; a1=y=x*(x^2-3*p); for(i=2, e, x=y*a1-p^3*a0; a0=a1; a1=x); a1)))))} /* Michael Somos Aug 23 2006 */

(PARI) {a(n)= if(n<0, 0, polcoeff( eta(x +x*O(x^n))^8, n))}

Cf. A033687.

Adjacent sequences: A000728 A000729 A000730 this_sequence A000732 A000733 A000734

Sequence in context: A082231 A029845 A124972 this_sequence A034433 A120081 A081963

sign,mult

njas