OFFSET
0,3
FORMULA
G.f. satisfies: A(x) = 1/2 + (1 + 2*x*A(x))/2 * Sum_{n>=0} (n+1)! * x^(2*n) * A(x)^n.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 41*x^5 + 128*x^6 + 409*x^7 +...
where:
A(x) = 1 + x*(1+x)*A(x)/(1+x*(1+x)*A(x)) + x^2*(1+2*x)^2*A(x)^2/(1+x*(1+2*x)*A(x))^2 + x^3*(1+3*x)^3*A(x)^3/(1+x*(1+3*x)*A(x))^3 + x^4*(1+4*x)^4*A(x)^4/(1+x*(1+4*x)*A(x))^4 +...
Also,
A(x) = 1/2 + (1 + 2*x*A(x))/2 * (1 + 2*x^2*A(x) + 6*x^4*A(x)^2 + 24*x^6*A(x)^3 + 120*x^8*A(x)^4 + 720*x^10*A(x)^5 + 5040*x^12*A(x)^6 +...).
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (x+m*x^2)^m*A^m / (1 + x*A+m*x^2*A +x*O(x^n))^m)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI)
{a(n)=local(A=1+x); for(i=1, n, A=1/2+(1+2*x*A)*sum(k=0, n, (k+1)!/2*x^(2*k)*(A+x*O(x^n))^k)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 25 2013
STATUS
approved