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A193382
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G.f. is the real part of the function C(x) that satisfies C(x) = 1 + x/C(I*x).
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2
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1, 1, 0, -1, -1, -1, 0, 2, 3, 3, 0, -6, -10, -10, 0, 20, 35, 35, 0, -70, -126, -126, 0, 252, 462, 462, 0, -924, -1716, -1716, 0, 3432, 6435, 6435, 0, -12870, -24310, -24310, 0, 48620, 92378, 92378, 0, -184756, -352716, -352716, 0, 705432, 1352078, 1352078, 0, -2704156
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OFFSET
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0,8
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LINKS
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FORMULA
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G.f.: real(C(x)) = (1+x)/2 + (1-x)/2 * sqrt(1 + 4*x^4)/(1 - 2*x + 2*x^2),
where C(x) = (1 + (1+I)*x)/2 + sqrt(1 + 4*x^4)/(2*(1 - (1-I)*x)),
and imag(C(x)) = (x/2)*(1 - sqrt(1 + 4*x^4)/(1 - 2*x + 2*x^2)).
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EXAMPLE
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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 +...
where
C(I*x)^-1 = 1 - I*x + (-1 - I)*x^2 - x^3 + (-1 + I)*x^4 + 2*I*x^5 +...
The real part of C(x) begins:
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 +...
The imaginary part of C(x) begins:
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 +...
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MATHEMATICA
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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] // Re]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 16 2017, adapted from PARI *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x/subst(A, x, I*x +x*O(x^n))); real(polcoeff(A, n))}
(PARI) {a(n)=polcoeff((1+x)/2+(1-x)/2*sqrt(1+4*x^4+x*O(x^n))/(1-2*x+2*x^2+x*O(x^n)), n)}
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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