%I A010816
%S A010816 1,3,0,5,0,0,7,0,0,0,9,0,0,0,0,11,0,0,0,0,0,13,0,0,0,0,0,0,15,0,0,0,0,
               0,
%T A010816 0,0,17,0,0,0,0,0,0,0,0,19,0,0,0,0,0,0,0,0,0,21,0,0,0,0,0,0,0,0,0,0,23,
%U A010816 0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0
%V A010816 1,-3,0,5,0,0,-7,0,0,0,9,0,0,0,0,-11,0,0,0,0,0,13,0,0,0,0,0,0,-15,0,0,
               0,0,0,0,0,17,0,0,
%W A010816 0,0,0,0,0,0,-19,0,0,0,0,0,0,0,0,0,21,0,0,0,0,0,0,0,0,0,0,-23,0,0,0,0,
               0,0,0,0,0,0,0,25,
%X A010816 0,0,0,0,0,0,0,0,0,0,0,0,-27,0,0,0,0,0,0,0
%N A010816 Expansion of Product_{k = 1 .. infinity} (1-x^k)^3.
%C A010816 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
%D A010816 M. Boylan, Exceptional congruences for the coefficients of certain eta-product 
               newforms, J. Number Theory 98 (2003), no. 2, 377-389.
%D A010816 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 
               2nd. ed. 1949, p. 117, Problem 22.
%D A010816 A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring 
               at most thrice, in preparation.
%D A010816 D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.4, Problem 23.
%D A010816 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.
%D A010816 S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 
               1957 Vol. 1, see page 267 MR0099904 (20 #6340)
%H A010816 S. R. Finch, <a href="http://arXiv.org/abs/math.NT/0701251">Powers of 
               Euler's q-Series</a>, (arXiv:math.NT/0701251).
%H A010816 <a href="Sindx_Pro.html#1mxtok">Index entries for expansions of Product_{k 
               >= 1} (1-x^k)^m</a>
%F A010816 Jacobi showed that Product_{k = 1 .. infinity} (1-x^k)^3 = Sum_{n=0 .. 
               infinity} (-1)^n (2n+1) x^(n*(n+1))/2.
%F A010816 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
%F A010816 Given g.f. A(x), then x*A(x)^8 is g.f. for A000594.
%F A010816 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
%F A010816 Expansion of f(-q)^3 in powers of q where f() is a Ramanujan theta function.
%F A010816 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
%F A010816 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
%F A010816 G.f.: Sum_{k>=0} (-1)^k * (2*k+1) * x^(k * (k+1) / 2).
%e A010816 q - 3*q^9 + 5*q^25 - 7*q^49 + 9*q^81 - 11*q^121 + 13*q^169 + ...
%o A010816 (PARI) a(n)=local(x); if(n<0, 0, if(issquare(8*n+1,&x),(-1)^(x\2)*x)) 
               /* Michael Somos Nov 08 2005 */
%o A010816 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x 
               + A)^3, n))}
%Y A010816 A116916(n) = a(3*n).
%Y A010816 Sequence in context: A052439 A143073 A154725 this_sequence A133089 A136599 
               A131986
%Y A010816 Adjacent sequences: A010813 A010814 A010815 this_sequence A010817 A010818 
               A010819
%K A010816 sign,easy,nice
%O A010816 0,2
%A A010816 N. J. A. Sloane (njas(AT)research.att.com).