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A216683 G.f. satisfies: A(x) = 1 + x*A(x) / ( A(I*x)*A(-I*x) ).

%I A216683 #14 Sep 14 2012 23:44:00
%S A216683 1,1,1,2,3,2,2,0,-5,-2,-2,8,38,20,20,-48,-269,-138,-138,392,2194,1132,
%T A216683 1132,-3344,-19010,-9812,-9812,30032,172332,89000,89000,-279136,
%U A216683 -1613629,-833626,-833626,2663432,15485978,8002172,8002172,-25938768,-151520246,-78309372,-78309372
%N A216683 G.f. satisfies: A(x) = 1 + x*A(x) / ( A(I*x)*A(-I*x) ).
%H A216683 Paul D. Hanna, <a href="/A216683/b216683.txt">Table of n, a(n) for n = 0..500</a>
%F A216683 G.f.: A(x) = 1/(1 - G(x^2)/x) where G(x) = x + G(-x)^2 is the g.f. of A143045.
%F A216683 a(4*n+1) = a(4*n+2) for n>=0.
%F A216683 Let A(x) = A0(x^4) + x*A1(x^4) + x^2*A2(x^2) + x^3*A3(x^4), then
%F A216683 (1) A1(x) = A2(x).
%F A216683 (2) A0(x) = A1(x) + x*A3(x) / (2*A0(x)).
%F A216683 (3) A0(x^4) - x^2*A2(x^4) = A(I*x)*A(-I*x).
%e A216683 G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 2*x^6 - 5*x^8 - 2*x^9 +...
%e A216683 Related expansions:
%e A216683 1/A(x) = 1 - x - x^3 + 2*x^5 + 3*x^7 - 10*x^9 - 18*x^11 + 68*x^13 + 131*x^15 - 530*x^17 - 1062*x^19 +...+ -A143045(n)*x^(2*n-1) +...
%e A216683 A(I*x)*A(-I*x) = 1 - x^2 + 3*x^4 - 2*x^6 - 5*x^8 + 2*x^10 + 38*x^12 - 20*x^14 - 269*x^16 + 138*x^18 + 2194*x^20 +...
%e A216683 The 4-sections of g.f. A(x) begin:
%e A216683 A0(x) = 1 + 3*x - 5*x^2 + 38*x^3 - 269*x^4 + 2194*x^5 - 19010*x^6 + 172332*x^7 +...
%e A216683 A1(x) = A2(x) = 1 + 2*x - 2*x^2 + 20*x^3 - 138*x^4 + 1132*x^5 - 9812*x^6 + 89000*x^7 +...
%e A216683 A3(x) = 2 + 8*x^2 - 48*x^3 + 392*x^4 - 3344*x^5 + 30032*x^6 - 279136*x^7 + 2663432*x^8 +...
%e A216683 where
%e A216683 A1(x) + x*A3(x)/(2*A0(x)) = 1 + 3*x - 5*x^2 + 38*x^3 - 269*x^4 + 2194*x^5 +...
%o A216683 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A/(subst(A, x, I*x+x*O(x^n))*subst(A, x, -I*x+x*O(x^n)))); polcoeff(A, n)}
%o A216683 for(n=0, 30, print1(a(n), ", "))
%Y A216683 Cf. A212527, A216681, A143045.
%K A216683 sign
%O A216683 0,4
%A A216683 _Paul D. Hanna_, Sep 14 2012