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G.f. is the imaginary part of the function C(x) that satisfies: C(x) = 1 + x/C(I*x).
2

%I #20 May 16 2017 11:44:12

%S -1,-1,0,1,2,2,0,-3,-6,-6,0,10,20,20,0,-35,-70,-70,0,126,252,252,0,

%T -462,-924,-924,0,1716,3432,3432,0,-6435,-12870,-12870,0,24310,48620,

%U 48620,0,-92378,-184756,-184756,0,352716,705432,705432,0,-1352078,-2704156,-2704156,0

%N G.f. is the imaginary part of the function C(x) that satisfies: C(x) = 1 + x/C(I*x).

%F G.f.: imag(C(x)) = (x/2)*(1 - sqrt(1 + 4*x^4)/(1 - 2*x + 2*x^2)),

%F where C(x) = (1 + (1+I)*x)/2 + sqrt(1 + 4*x^4)/(2*(1 - (1-I)*x)),

%F and real(C(x)) = (1+x)/2 + (1-x)/2 * sqrt(1 + 4*x^4)/(1 - 2*x + 2*x^2).

%e C(x) = 1 + x - I*x^2 + (-1 - I)*x^3 - x^4 + (-1 + I)*x^5 + 2*I*x^6 + (2 + 2*I)*x^7 + 3*x^8 + (3 - 3*I)*x^9 - 6*I*x^10 + (-6 - 6*I)*x^11 - 10*x^12 +...

%e where

%e C(I*x)^-1 = 1 - I*x + (-1 - I)*x^2 - x^3 + (-1 + I)*x^4 + 2*I*x^5 +...

%e The real part of C(x) begins:

%e real(C(x)) = 1 + x - x^3 - x^4 - x^5 + 2*x^7 + 3*x^8 + 3*x^9 - 6*x^11 - 10*x^12 - 10*x^13 + 20*x^15 + 35*x^16 + 35*x^17 - 70*x^19 - 126*x^20 - 126*x^21 +...

%e The imaginary part of C(x) begins:

%e imag(C(x)) = -x^2 - x^3 + x^5 + 2*x^6 + 2*x^7 - 3*x^9 - 6*x^10 - 6*x^11 + 10*x^13 + 20*x^14 + 20*x^15 - 35*x^17 - 70*x^18 - 70*x^19 + 126*x^21 +...

%t a[n_] := Module[{A = 1 + x}, For[i = 1, i <= n, i++, A = 1 + x/(A /. x -> I*x + x*O[x]^n)]; Coefficient[A, x, n] // Im]; Table[a[n], {n, 2, 60}] (* _Jean-François Alcover_, May 16 2017, adapted from PARI *)

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x/subst(A, x, I*x +x*O(x^n))); imag(polcoeff(A, n))}

%o (PARI) {a(n)=polcoeff((x/2)*(1-sqrt(1+4*x^4+x*O(x^n))/(1-2*x+2*x^2+x*O(x^n))),n)}

%Y Cf. A193382 (real), A193384 (norm).

%K sign

%O 2,5

%A _Paul D. Hanna_, Jul 24 2011