OFFSET
0,3
COMMENTS
The limit a(n+1)/a(n) seems to exist, with a value near 2.81...
FORMULA
G.f. satisfies: (A(x) + A(-x))/2 = 1 + x*Sum_{n>=0} b(n)*b(n+1)*x^(2n+1) and
(A(x) - A(-x))/2 = x*Sum_{n>=0} b(n)^2*x^(2n) where Sum_{n>=0} b(n)*x^n = A(x)^2 with a(0)=1.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 60*x^6 +...
The square of the g.f. begins:
A(x)^2 = 1 + 2*x + 5*x^2 + 12*x^3 + 32*x^4 + 86*x^5 +...+ b(n)*x^n +...
where the odd bisection yields the series with squared coefficients:
(A(x) - A(-x))/2 = x + 4*x^3 + 25*x^5 + 144*x^7 + 1024*x^9 +...+ b(n)^2*x^(2n+1) +...
and the even bisection yields the series:
(A(x) + A(-x))/2 = 1 + 2*x^2 + 10*x^4 + 60*x^6 + 384*x^8 +...+ b(n)*b(n+1)*x^(2n+2) +...
PROG
(PARI) {a(n)=local(A=1+x, B); for(i=1, n, B=(A+x*O(x^n))^2; A=1+x*sum(m=0, n\2, polcoeff(B, m)*polcoeff(B, m+1)*x^(2*m+1)) +x*sum(m=0, n\2, polcoeff(B, m)^2*x^(2*m))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 22 2010
STATUS
approved