%I A002107 M0091 N0028
%S A002107 1,2,1,2,1,2,2,0,2,2,1,0,0,2,3,2,2,0,0,2,2,0,0,2,1,0,2,2,2,2,1,2,0,2,2,
%T A002107 2,2,0,2,0,4,0,0,0,1,2,0,0,2,0,2,2,1,2,0,2,2,0,0,2,0,2,0,2,2,0,4,0,0,2,
%U A002107 1,2,0,2,0,0,0,2,2,4,1,0,0,2,2,2,2,0,0,2,0,2,0,2,2,0,2,0,0,0,2,2,1,2,2
%V A002107 1,-2,-1,2,1,2,-2,0,-2,-2,1,0,0,2,3,-2,2,0,0,-2,-2,0,0,-2,-1,0,2,2,-2,
               2,1,2,0,2,-2,-2,
%W A002107 2,0,-2,0,-4,0,0,0,1,-2,0,0,2,0,2,2,1,-2,0,2,2,0,0,-2,0,-2,0,-2,2,0,-4,
               0,0,-2,-1,2,0,2,
%X A002107 0,0,0,-2,2,4,1,0,0,2,-2,2,-2,0,0,2,0,-2,0,-2,-2,0,-2,0,0,0,2,-2,-1,-2,
               -2
%N A002107 Expansion of Product (1-x^k)^2, k=1..inf.
%C A002107 Number of partitions of n into an even number of distinct parts - partitions 
               of n into an odd number of distinct parts, with 2 types of each part. 
               E.g. for n=4, we consider k and k* to be different versions of k 
               and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*. The even 
               partitions number 5 and the odd partitions number 4, so a(4)=5-4=1 
               - Jon Perry (perry(AT)globalnet.co.uk), Apr 04 2004
%C A002107 Also, number of different partitions of n into parts of -2 different 
               kinds (based upon formal analogy) - Michele Dondi (blazar(AT)lcm.mi.infn.it), 
               Jun 29 2004
%D A002107 M. Boylan, Exceptional congruences for the coefficients of certain eta-product 
               newforms, J. Number Theory 98 (2003), no. 2, 377-389.
%D A002107 J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. 
               Soc., 21 (1889), 182-194.
%D A002107 M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
%D A002107 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002107 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A002107 T. D. Noe, <a href="b002107.txt">Table of n, a(n) for n=0..1000</a>
%H A002107 S. R. Finch, <a href="http://arXiv.org/abs/math.NT/0701251">Powers of 
               Euler's q-Series</a>, (arXiv:math.NT/0701251).
%H A002107 <a href="Sindx_Pro.html#1mxtok">Index entries for expansions of Product_{k 
               >= 1} (1-x^k)^m</a>
%F A002107 a(n)=b(12n+1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, 
               b(p^e) = (1+(-1)^e)/2 if p == 7, 11 (mod 12), b(p^e) = (-1)^(e/2)(1+(-1)^e)/
               2 if p == 5 (mod 12), b(p^e) = (e+1)*(-1)^(e*x) if p == 1 (mod 12) 
               and p = x^2+9y^2. - Michael Somos Sep 16 2006
%o A002107 (PARI) {a(n)=local(A, p, e, x); if(n<0, 0, n=12*n+1; A=factor(n); prod(k=1, 
               matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p<5, 0, if(p%12>1, if(e%2, 
               0, (-1)^((p%12==5)*e/2)), for(i=1, sqrtint(p\9), if(issquare(p-9*i^2), 
               x=i; break)); (e+1)*(-1)^(e*x))))))} /* Michael Somos Aug 30 2006 
               */
%o A002107 (PARI) {a(n)=if(n<0, 0, polcoeff( eta(x+x*O(x^n))^2, n))} /* Michael 
               Somos Aug 30 2006 */
%Y A002107 Cf. A000712 (reciprocal of g.f.).
%Y A002107 Sequence in context: A063279 A124333 A144757 this_sequence A133099 A006571 
               A094781
%Y A002107 Adjacent sequences: A002104 A002105 A002106 this_sequence A002108 A002109 
               A002110
%K A002107 sign,nice
%O A002107 0,2
%A A002107 N. J. A. Sloane (njas(AT)research.att.com).