%I A001938 M3475 N1412
%S A001938 1,4,14,40,101,236,518,1080,2162,4180,7840,14328,25591,44776,76918,129952,
%T A001938 216240,354864,574958,920600,1457946,2285452,3548550,5460592,8332425,12614088,
%U A001938 18953310,28276968,41904208,61702876,90304598,131399624
%V A001938 1,-4,14,-40,101,-236,518,-1080,2162,-4180,7840,-14328,25591,-44776,76918,
               -129952,
%W A001938 216240,-354864,574958,-920600,1457946,-2285452,3548550,-5460592,8332425,
               -12614088,
%X A001938 18953310,-28276968,41904208,-61702876,90304598,-131399624
%N A001938 Expansion of k/(4q^(1/2)) in powers of q, where k is the elliptic function 
               defined by sqrt(k) = theta_2/theta_3.
%D A001938 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001938 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001938 A. Cayley, A memoir on the transformation of elliptic functions, Collected 
               Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, 
               Vol. 9, p. 128.
%D A001938 E. T. Copson, An Introduction to the Theory of Functions of a Complex 
               Variable, 1935, Oxford University Press, p. 385.
%D A001938 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; Eq. (34.3).
%H A001938 T. D. Noe, <a href="b001938.txt">Table of n, a(n) for n=0..1000</a>
%F A001938 G.f.: (Product_{n>0} (1+x^(2n))/(1+x^(2n-1)))^4.
%F A001938 G.f. A(x) satisfies 1=(1-16xA(x)^2)(1+16xA(-x)^2). - Michael Somos, Mar 
               26 2004
%F A001938 Given g.f. A(x), then B(x)=A(x^2)x satisfies 0=f(B(x), B(x^2)) where 
               f(u, v)=v-(u(1+4v))^2. - Michael Somos, Mar 26 2004
%F A001938 Expansion of q^(-1/2)* (eta(q)* eta(q^4)^2/ eta(q^2)^3)^4 in powers of 
               q.
%F A001938 Euler transform of period 4 sequence [ -4, 8, -4, 0, ...].
%o A001938 (PARI) {a(n)=local(A,A2,m); if(n<0, 0, n=2*n+1; A=x+O(x^3); m=2; while(m<n, 
               m*=2; A=subst(A,x,x^2); A=sqrt(A)/(1+4*A)); polcoeff(A,n))} - Michael 
               Somos, Mar 26 2004
%o A001938 (PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( (eta(x+A)* 
               eta(x^4+A)^2/ eta(x^2+A)^3)^4, n))} - Michael Somos, Mar 26 2004
%Y A001938 Cf. A127931, A127932, A127393, A079006, A001936.
%Y A001938 a(n) = (-1)^n* A093160(n).
%Y A001938 Sequence in context: A144141 A066375 A093160 this_sequence A066368 A121593 
               A023003
%Y A001938 Adjacent sequences: A001935 A001936 A001937 this_sequence A001939 A001940 
               A001941
%K A001938 sign,nice
%O A001938 0,2
%A A001938 N. J. A. Sloane (njas(AT)research.att.com).
%E A001938 Edited by N. J. A. Sloane (njas(AT)research.att.com), Mar 31 2007