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Norm of coefficients in g.f. C(x) that satisfies: C(x) = 1 + x/C(I*x).
2

%I #10 May 16 2017 11:44:00

%S 1,1,1,2,1,2,4,8,9,18,36,72,100,200,400,800,1225,2450,4900,9800,15876,

%T 31752,63504,127008,213444,426888,853776,1707552,2944656,5889312,

%U 11778624,23557248,41409225,82818450,165636900,331273800,590976100,1181952200,2363904400

%N Norm of coefficients in g.f. C(x) that satisfies: C(x) = 1 + x/C(I*x).

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

%e G.f.: 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 the g.f. 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 the g.f. 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)]; Norm[Coefficient[A, x, n]]^2]; Table[a[n], {n, 0, 40}] (* _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))); norm(polcoeff(A, n))}

%Y Cf. A193382 (real), A193383 (imag).

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jul 24 2011