A216681 - OEIS

%I #13 Sep 14 2012 17:32:46

%S 1,1,3,17,85,333,1883,13153,76329,363033,2304867,17067553,104686957,

%T 534812789,3558451915,27086552833,170930393745,906063493617,

%U 6183676880195,47831931663921,307091159448965,1664876216837789,11545009017568635,90248125157828449

%N G.f.: A(x) = 1 + x*A(x)^3 / ( A(I*x)*A(-I*x) ), where I^2 = -1.

%e G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 85*x^4 + 333*x^5 + 1883*x^6 +...

%e Related expansions:

%e A(x)^2 = 1 + 2*x + 7*x^2 + 40*x^3 + 213*x^4 + 938*x^5 + 5231*x^6 +...

%e A(x)^3 = 1 + 3*x + 12*x^2 + 70*x^3 + 393*x^4 + 1893*x^5 + 10632*x^6 +...

%e A(x)*A(-x) = 1 + 5*x^2 + 145*x^4 + 3321*x^6 + 133553*x^8 + 4103661*x^10 +...

%e A(I*x)*A(-I*x) = 1 - 5*x^2 + 145*x^4 - 3321*x^6 + 133553*x^8 - 4103661*x^10 + 184486609*x^12 - 6359604209*x^14 + 302240850145*x^16 - 11073953305621*x^18 +...

%e Note that a bisection of 1/A(x)^2 equals a bisection of 1/A(x)^3:

%e 1/A(x)^2 = 1 - 2*x - 3*x^2 - 20*x^3 - 72*x^4 - 108*x^5 - 1196*x^6 +...

%e 1/A(x)^3 = 1 - 3*x - 3*x^2 - 25*x^3 - 72*x^4 + 12*x^5 - 1196*x^6 +...

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^3/(subst(A, x, I*x+x*O(x^n))*subst(A, x, -I*x+x*O(x^n)))); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A212527, A216712, A216713.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 14 2012