%I #6 Mar 30 2012 18:37:33
%S 1,1,2,4,10,28,84,260,818,2609,8434,27622,91524,306270,1033364,
%T 3510960,12000898,41239249,142387668,493732424,1718653462,6003501630,
%U 21037928794,73937288814,260544327156,920376228329,3258619058158,11561493797242,41100002807572
%N G.f. satisfies: A(x) = 1 + (eta(x^2*A(x)^2)^10 / (eta(x*A(x))^4 * eta(x^4*A(x)^4)^4) - 1)/4, where eta(q) is the Dedekind eta function without the q^(1/24) factor.
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F (1) A(x) = 1 + Sum_{n>=1} (-1)^[n/2] * (x*A(x))^(n*(n+1)/2) / (1 + (-x*A(x))^n);
%F (2) A(x) = 1 + Sum_{n>=1} x^n*A(x)^n / (1 + x^(2*n)*A(x)^(2*n));
%F (3) A(x) = 1 + Sum_{n>=1} (-1)^(n-1)*x^(2*n-1)*A(x)^(2*n-1) / (1 + x^(2*n-1)*A(x)^(2*n-1)).
%F Let G(x) = 1 + g(x) where g(x) is the g.f. of A002654, then:
%F (4) A(x) = G(x*A(x)) where G(x) = A(x/G(x));
%F (5) A(x) = (1/x)*Series_Reversion( x/G(x));
%F (6) a(n) = [x^n] G(x)^(n+1) / (n+1);
%F where A002654(n) is the number of ways of writing n as a sum of at most two nonzero squares (with respect to order).
%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 28*x^5 + 84*x^6 +...
%e where the g.f. A = A(x) satisfies the series identities:
%e (1) A(x) = 1 + x*A/(1 - x*A) - x^3*A^3/(1 + x^2*A^2) - x^6*A^6/(1 - x^3*A^3) + x^10*A^10/(1 + x^4*A^4) + x^15*A^15/(1 - x^5*A^5) +...
%e (2) A(x) = 1 + x*A/(1 + x^2*A^2) + x^2*A^2/(1 + x^4*A^4) + x^3*A^3/(1 + x^6*A^6) + x^4*A^4/(1 + x^8*A^8) +...
%e (3) A(x) = 1 + x*A/(1 - x*A) - x^3*A^3/(1 - x^3*A^3) + x^5*A^5/(1 - x^5*A^5) - x^7*A^7/(1 - x^7*A^7) +...
%e (4) G.f. A(x) satisfies A(x) = G(x*A(x)) where G(x) = 1 + g.f. of A002654:
%e A(x) = 1 + x*A(x) + x^2*A(x)^2 + x^4*A(x)^4 + 2*x^5*A(x)^5 + x^8*A(x)^8 + x^9*A(x)^9 + 2*x^10*A(x)^10 +...+ A002654(n)*x^n*A(x)^n +...
%e (6) Coefficients in the powers of G(x) = 1 + g.f. of A002654 begin:
%e G^1: [(1), 1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 0, 2, 0, 0, 1, 2, ...];
%e G^2: [1,(2), 3, 2, 3, 6, 6, 4, 3, 8, 12, 6, 6, 10, 12, 12, 3, ...];
%e G^3: [1, 3, (6), 7, 9, 15, 22, 24, 21, 30, 48, 51, 46, 51, ...];
%e G^4: [1, 4, 10,(16), 23, 36, 58, 80, 91, 112, 162, 216, ...];
%e G^5: [1, 5, 15, 30,(50), 81, 135, 210, 285, 370, 511, 720, ...];
%e G^6: [1, 6, 21, 50, 96, (168), 291, 486, 741, 1052, ...];
%e G^7: [1, 7, 28, 77, 168, 322, (588), 1037, 1715, 2646, ...];
%e G^8: [1, 8, 36, 112, 274, 576, 1120,(2080), 3663, 6064, ...];
%e G^9: [1, 9, 45, 156, 423, 972, 2022, 3960, (7362), 12940, ...]; ...
%e where the coefficients in parenthesis form the initial terms of this sequence:
%e A = [1/1, 2/2, 6/3, 16/4, 50/5, 168/6, 588/7, 2080/8, 7362/9, ...].
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+(eta(x^2*A^2)^10/(eta(x*A)^4*eta(x^4*A^4)^4)-1)/4+x*O(x^n));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+sum(m=1,n,(-1)^(m\2)*(x*A)^(m*(m+1)/2)/(1+(-x*A+x*O(x^n))^m)));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^m/(1+x^(2*m)*(A+x*O(x^n))^(2*m))));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+sum(m=1,n,(-1)^(m-1)*x^(2*m-1)*(A+x*O(x^n))^(2*m-1)/(1-x^(2*m-1)*(A+x*O(x^n))^(2*m-1))));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x,G=1+sum(m=1,n\2+1,(-1)^(m-1)*x^(2*m-1)/(1-x^(2*m-1)+x^2*O(x^n))));A=1/x*serreverse(x/G);polcoeff(A,n)}
%Y Cf. A002654.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 11 2011