%I A081362
%S A081362 1,1,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,5,5,6,7,8,8,9,11,12,12,14,16,17,18,
%T A081362 20,23,25,26,29,33,35,37,41,46,49,52,57,63,68,72,78,87,93,98,107,117,
%U A081362 125,133,144,157,168,178,192,209,223,236,255,276,294,312,335,361,385
%V A081362 1,-1,0,-1,1,-1,1,-1,2,-2,2,-2,3,-3,3,-4,5,-5,5,-6,7,-8,8,-9,11,-12,12,
               -14,16,-17,18,
%W A081362 -20,23,-25,26,-29,33,-35,37,-41,46,-49,52,-57,63,-68,72,-78,87,-93,98,
               -107,117,
%X A081362 -125,133,-144,157,-168,178,-192,209,-223,236,-255,276,-294,312,-335,361,
               -385
%N A081362 Expansion of q^(1/24) * eta(q) / eta(q^2) in powers of q.
%C A081362 (Number of partitions of n into an even number of parts) - (number of 
               partitions of n into an odd number of parts). [Fine]
%D A081362 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 58, Eq. (26.56) [essentially the function phi(q)]; 
               also p. 38, Eq. (22.14).
%F A081362 G.f.: Product_{k>0} (1-x^(2k-1)) = Product_{k>0} 1/(1+x^k) = 1+Sum_{k>
               0} (-x)^k/(Product_{i=1..k} (1-x^i)).
%F A081362 This is the convolution inverse of A000009 (partitions into distinct 
               parts) - i.e. the negation of the INVERTi transform of A000009. - 
               Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 06 2006
%F A081362 Expansion of chi(-q) in powers of q where chi() is a Ramanujan theta 
               function.
%F A081362 Given g.f. A(x), then B(x)=A(x^3)^8/x satisfies 0=f(B(x), B(x^2)) where 
               f(u, v)=u^2*v +16*u -v^2.
%F A081362 G.f. A(x) satisfies A(x^2)=A(x)A(-x).
%F A081362 Euler transform of period 2 sequence [ -1, 0, ...].
%F A081362 Expansion of q^(1/24) f1(t) in powers of q = exp(Pi i t) where f1() is 
               a Weber function.
%e A081362 q^-1 - q^23 - q^71 + q^95 - q^119 + q^143 - q^167 + 2*q^191 - 2*q^215 
               + ...
%o A081362 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)/eta(x^2+A), 
               n))}
%Y A081362 A000700(n)=(-1)^n a(n).
%Y A081362 Sequence in context: A035435 A025775 A000700 this_sequence A112216 A058688 
               A132322
%Y A081362 Adjacent sequences: A081359 A081360 A081361 this_sequence A081363 A081364 
               A081365
%K A081362 sign
%O A081362 0,9
%A A081362 Michael Somos, Mar 18 2003