OFFSET
1,3
COMMENTS
a(n) = n * Sum_{k=0..n-1} (-1)^(n-k-1)*A122941(n-k,k)/(n-k).
FORMULA
L.g.f.: A(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=0} log(1 + F_n(x))/2^(n+1) where F_0(x)=x, F_{n+1}(x) = F_n(x+x^2); a sum involving self-compositions of x+x^2 (cf. A122888).
EXAMPLE
To illustrate A(x+x^2) = 2*A(x) - log(1+x):
A(x) = x + 1*x^2/2 + 4*x^3/3 + 17*x^4/4 + 106*x^5/5 + 796*x^6/6 +...
A(x+x^2) = x + 3*x^2/2 + 7*x^3/3 + 35*x^4/4 + 211*x^5/5 + 1593*x^6/6 +...
PROG
(PARI) {a(n)=local(A=x+x*O(x^n)); for(i=0, n, A=-A+subst(A, x, x+x^2)+log(1+x+x*O(x^n))); n*polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 25 2006
STATUS
approved