OFFSET
0,4
COMMENTS
Compare to: F(x) = exp( Integral G(x) dx ) such that G(x) = exp(1-exp(x)) * exp( Integral F(x) dx ) holds when F(x) = exp(x).
FORMULA
Logarithmic derivative of A268170.
EXAMPLE
E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 9*x^4/4! + 46*x^5/5! + 245*x^6/6! + 1474*x^7/7! + 10315*x^8/8! + 82174*x^9/9! + 726591*x^10/10! + 7038632*x^11/11! + 74216949*x^12/12! +...
where
C(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 65*x^5/5! + 326*x^6/6! + 1947*x^7/7! + 13410*x^8/8! + 104181*x^9/9! + 900214*x^10/10! + 8566655*x^11/11! +...+ A268170(n)*x^n/n! +...
and A(x) and C(x) satisfy:
(1) A(x) = C'(x)/C(x),
(2) C(x) = A'(x)/A(x) + exp(x) - 1,
(3) log(C(x)) = Integral A(x) dx,
(4) log(A(x)) = Integral C(x) dx + 1+x - exp(x).
PROG
(PARI) {a(n) = my(A=1+x, C=1+x); for(i=0, n, A = exp(1+x - exp(x +x*O(x^n))) * exp( intformal( C + x*O(x^n) ) ); C = exp( intformal( A ) ); ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 29 2016
STATUS
approved