OFFSET
1,3
COMMENTS
Higher derivatives of the e.g.f. involve nested exponential functions, e.g.:
A''(x)*exp(-x)/(1+x) = 1 + A( x*exp(x)*exp( x*exp(x) ) ).
FORMULA
a(n) = Sum_{k=1..n-1} C(n-1,k)* k^(n-k-1)* a(k) for n>1 with a(1)=1.
EXAMPLE
E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 12*x^4/4! + 76*x^5/5! +...
Related expansions.
A'(x) = 1 + x + 3*x^2/2! + 12*x^3/3! + 76*x^4/4! + 671*x^5/5! +...
A(x*exp(x)) = x + 3*x^2/2! + 12*x^3/3! + 76*x^4/4! + 671*x^5/5! +...
MATHEMATICA
terms = 19; B[_] = 0; Do[B[x_] = 1 + Integrate[B[x], x] /. x -> x Exp[x] + O[x]^terms // Normal, terms];
A[x_] = Integrate[B[x], x];
CoefficientList[A[x]/x, x] * Range[terms]! (* Jean-François Alcover, Sep 15 2011, updated Jan 11 2018 *)
PROG
(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x+intformal(subst(A, x, x*exp(x+1*O(x^n))))); n!*polcoeff(A, n)}
(PARI) {a(n)=if(n<1, 0, if(n==1, 1, sum(k=1, n-1, binomial(n-1, k)*k^(n-k-1)*a(k))))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 17 2011
STATUS
approved