OFFSET
1,3
FORMULA
G.f. satisfies: A(x) = Integral F(x) dx where F(x) is the g.f. of A211195 such that F(x) = 1 + 2*A(x*G(x)) and G(x) = F(x*G(x)). - Paul D. Hanna, Nov 21 2013
EXAMPLE
G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 48*x^5 + 372*x^6 + 3440*x^7 +...
The derivative of g.f. A(x) equals F(x), which is the g.f. of A211195 and begins:
F(x) = 1 + 2*x + 6*x^2 + 32*x^3 + 240*x^4 + 2232*x^5 + 24080*x^6 +...
where F(x) = 1 + 2*A(x*G(x)) and G(x) = F(x*G(x)):
G(x) = 1 + 2*x + 10*x^2 + 76*x^3 + 728*x^4 + 8104*x^5 + 100520*x^6 +...
PROG
(PARI) a(n)=local(F=1+2*x+sum(j=2, n-2, (j+1)*a(j+1)*x^j)+x*O(x^n)); if(n<1, 0, if(n==1, 1, 2*sum(k=0, n-2, polcoeff(F^(n-k-1), k)*polcoeff(F^(k+1)/(k+1), n-k-2))/n))
for(n=1, 25, print1(a(n), ", "))
(PARI) a(n)=local(B=1+x); for(i=1, n, B=1+2*subst(intformal(B), x, serreverse(x/B +x*O(x^n)))); polcoeff(intformal(B), n)
for(n=1, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 21 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 03 2013
EXTENSIONS
Offset changed to 1 (from 0) by Paul D. Hanna, Nov 21 2013
STATUS
approved