0,2

Also, number of different partitions of n into parts of -3 different kinds (based upon formal analogy) - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004

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

T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 117, Problem 22.

A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.

D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.4, Problem 23.

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

S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 267 MR0099904 (20 #6340)

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

Jacobi showed that Product_{k = 1 .. infinity} (1-x^k)^3 = Sum_{n=0 .. infinity} (-1)^n (2n+1) x^(n*(n+1))/2.

Given g.f. A(x), then q*A(q^8) = eta(q^8)^3 = theta_2(q^4)theta_3(q^4)theta_4(q^4)/2 = theta_1'(q^4)/(2pi). - Michael somos Nov 08 2005

Given g.f. A(x), then x*A(x)^8 is g.f. for A000594.

a(n)=b(8n+1) where b(n) is multiplicative and b(p^e) = 0 if e odd, b(2^e) = 0^e, b(p^e) = p^(e/2) if p == 1 (mod 4), b(p^e) = (-p)^(e/2) if p == 3 (mod 4). - Michael Somos Aug 22 2006

Expansion of f(-q)^3 in powers of q where f() is a Ramanujan theta function.

G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 2^(9/2) (t/i)^(3/2) f(t) where q = exp(2 pi i t). - Michael Somos Sep 09 2007

a(3*n+2) = a(5*n+2) = a(5*n+4) = a(9*n+4) = a(9*n+7) = 0. a(9*n+1) = -3 * a(n). a(25*n+3) = 5 * a(n). - Michael Somos Sep 09 2007

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

q - 3*q^9 + 5*q^25 - 7*q^49 + 9*q^81 - 11*q^121 + 13*q^169 + ...

(PARI) a(n)=local(x); if(n<0, 0, if(issquare(8*n+1, &x), (-1)^(x\2)*x)) /* Michael Somos Nov 08 2005 */

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

A116916(n) = a(3*n).

Adjacent sequences: A010813 A010814 A010815 this_sequence A010817 A010818 A010819

Sequence in context: A052439 A143073 A154725 this_sequence A133089 A136599 A131986

sign,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).