A193383 G.f. is the imaginary part of the function C(x) that satisfies: C(x) = 1 + x/C(I*x).

-1, -1, 0, 1, 2, 2, 0, -3, -6, -6, 0, 10, 20, 20, 0, -35, -70, -70, 0, 126, 252, 252, 0, -462, -924, -924, 0, 1716, 3432, 3432, 0, -6435, -12870, -12870, 0, 24310, 48620, 48620, 0, -92378, -184756, -184756, 0, 352716, 705432, 705432, 0, -1352078, -2704156, -2704156, 0

Offset

2,5

Links

Table of n, a(n) for n=2..52.

Formula

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

Example

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 +...

Mathematica

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 *)

Prog

(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))}
(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)}

Crossrefs

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

Sequence in context: A216973 A061314 A366730 * A218033 A326500 A255903

Adjacent sequences: A193380 A193381 A193382 * A193384 A193385 A193386

Keyword

sign

Author

Paul D. Hanna, Jul 24 2011

Status

approved