%I A083365
%S A083365 1,1,2,3,4,6,9,12,16,22,29,38,50,64,82,105,132,166,208,258,320,395,484,
%T A083365 592,722,876,1060,1280,1539,1846,2210,2636,3138,3728,4416,5222,6163,
%U A083365 7256,8528,10006,11716,13696,15986,18624,21666,25169,29190,33808,39104
%V A083365 1,-1,2,-3,4,-6,9,-12,16,-22,29,-38,50,-64,82,-105,132,-166,208,-258,320,
               -395,484,
%W A083365 -592,722,-876,1060,-1280,1539,-1846,2210,-2636,3138,-3728,4416,-5222,
               6163,
%X A083365 -7256,8528,-10006,11716,-13696,15986,-18624,21666,-25169,29190,-33808,
               39104
%N A083365 Expansion of q^(-1/8)eta(q)eta(q^4)^2/eta(q^2)^3 in powers of q.
%D A083365 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 
               221 Entry 1(i).
%D A083365 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 A083365 H. T. Davis, Introduction to nonlinear differential and integral equations, 
               Dover Publications, Inc., New York 1962, p. 170 MR0181773 (31 #6000)
%D A083365 W. Duke, Continued fractions and modular functions, Bull. Amer. Math. 
               Soc., 42 (2005), 137-162; see Eqs. (9.1),(9.3).
%F A083365 Given g.f. A(x), then B(x)=x*A(x^8) satisfies 0=f(B(x), B(x^2)) where 
               f(u, v)=v^2-u^4(1+4v^4).
%F A083365 Given g.f. A(x), then B(x)=x*A(x^8) satisfies 0=f(B(x), B(x^3)) where 
               f(u, v)=v^4-u^4+uv+4(uv)^3.
%F A083365 Given g.f. A(x), then B(x)=x*A(x^8) satisfies 0=f(B(x), B(x^2), B(x^4)) 
               where f(u, v, w)=w-u^2v(1+2w^2) . Michael Somos May 29 2005
%F A083365 Given g.f. A(x), then B(x)=x*A(x^8) satisfies 0=f(B(x), B(x^2), B(x^3), 
               B(x^6)) where f(u1, u2, u3, u6)=u2*u6-u1*u3*(u2^2+u6^2) . - Michael 
               Somos May 29 2005
%F A083365 Given g.f. A(x), then B(x)=sqrt(2)*x*A(x^8) satisfies 0=f(B(x), B(x^7)) 
               where f(u, v)=(1-u^8)(1-v^8)-(1-uv)^8 . - Michael Somos Jan 01 2006
%F A083365 G.f.: Product_{n>0} (1+x^(2n))/(1+x^(2n-1)) = (Sum_{k>0} x^(k^2-k))/(Sum_{k>
               0} x^((k^2-k)/2)).
%F A083365 Expansion of f(q) / f(-q^4) = phi(q) / psi(q) = psi(q) / psi(q^2) = phi(-q^2) 
               / psi(-q) = chi(q) * chi(-q^2) = chi^2(q) * chi(-q) = chi^2(-q^2) 
               / chi(-q) = (phi(q) / psi(q^2))^(1/2) in powers of q where phi(), 
               psi(), chi(), f() are Ramanujan theta functions.
%F A083365 Expansion of k^(1/4) / (2^(1/2) * q^(1/8)) in powers of q where k is 
               elliptic modulus and q is the nome.
%F A083365 Euler transform of period 4 sequence [ -1, 2, -1, 0, ...].
%F A083365 G.f.: 1 / (1 + x / (1 + x + x^2 / (1 + x^2 + x^3 / (1 + x^3 + ...)))).
%e A083365 q - q^9 + 2*q^17 - 3*q^25 + 4*q^33 - 6*q^41 + 9*q^49 - 12*q^57 + 16*q^65 
               + ...
%o A083365 (PARI) a(n)=local(A,m); if(n<0,0,A=1+O(x); m=1; while(m<=n,m*=2; A=subst(A,
               x,x^2); A=sqrt(A/(1+4*x*A^2))); polcoeff(sqrt(A),n))
%o A083365 (PARI) {a(n)=local(A); if(n<0, 0, A=contfracpnqn(matrix(2, (sqrtint(8*n+1)+1)\2, 
               i, j, if(i==1, x^(j-1), 1+if(j>1, x^(j-1))))); polcoeff(A[2,1]/A[1,
               1]+x*O(x^n), n))}
%o A083365 (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, n))}
%Y A083365 A001935(n)=(-1)^n a(n). Convolution square is A079006.
%Y A083365 Sequence in context: A073576 A069907 A001935 this_sequence A007604 A013950 
               A018550
%Y A083365 Adjacent sequences: A083362 A083363 A083364 this_sequence A083366 A083367 
               A083368
%K A083365 sign
%O A083365 0,3
%A A083365 Michael Somos, Apr 24 2003