OFFSET
0,3
FORMULA
G.f. satisfies: A(x) = 1 + A(x)^2 * Series_Reversion(x*A(-x)).
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 87*x^5 + 323*x^6 + 1213*x^7 +...
where
A(x*A(-x)) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 1430*x^8 +...+ A000108(n)*x^n +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 22*x^3 + 75*x^4 + 272*x^5 + 1034*x^6 +...
Series_Reversion(x*A(-x)) = x + x^2 - x^3 - 2*x^4 + x^5 + 10*x^6 - x^7 - 80*x^8 - 35*x^9 + 841*x^10 + 956*x^11 +...
PROG
(PARI) /* From formula: A(x*A(-x)) = C(x) = 1 + x*C(x)^2 */
{a(n)=local(A=1+x, C=sum(m=0, n, binomial(2*m, m)/(m+1)*x^m));
for(i=1, n, A=subst(C, x, serreverse(x*subst(A, x, -x) +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* A(x) = 1 + A(x)^2*Series_Reversion(x*A(-x)): */
{a(n)=local(A=1+x); for(i=1, n, A=1+A^2*serreverse(x*subst(A, x, -x) +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 24 2014
STATUS
approved