OFFSET
0,4
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.
(1) A(x) = log(1 - (1/2)*log(1-2*x)).
(2) a(n) = (-1)^(n-1) * Sum_{k=1..n} 2^(n-k) * (k-1)! * Stirling1(n, k) for n > 0.
(3) a(n) = 2^(n-1)*(n-1)! - Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * 2^(k-1) * a(n-k) for n > 0.
EXAMPLE
E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 22*x^4/4! + 168*x^5/5! + 1616*x^6/6! + 18800*x^7/7! + 256432*x^8/8! + 4012288*x^9/9! + ...
where
exp(A(x)) = 1 + x + 2*x^2/2 + 4*x^3/3 + 8*x^4/4 + 16*x^5/5 + ... + 2^(n-1)*x^n/n + ...
PROG
(PARI) {a(n) = n!*polcoeff( log((1 - log(sqrt(1-2*x +x*O(x^n))))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = (-1)^(n-1) * sum(k=1, n, 2^(n-k) * (k-1)! * stirling(n, k, 1) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = if (n<1, 0, 2^(n-1)*(n-1)! - sum(k=1, n-1, binomial(n-1, k)*(k-1)! * 2^(k-1) * a(n-k)))}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 09 2023
STATUS
approved