OFFSET
0,4
FORMULA
E.g.f. A(x) satisfies:
(1) A(x) = 1 + Series_Reversion( 1/A(x) * Integral A(x) dx ).
(2) Integral A(x) dx = A(x) * Series_Reversion( A(x) - 1 ).
(3) A(x) = exp( Integral (1 - B'(x))/B(x) dx ), where B(A(x) - 1) = x.
EXAMPLE
E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 9*x^4/4! + 76*x^5/5! + 980*x^6/6! + 17304*x^7/7! + 393463*x^8/8! + 11072376*x^9/9! + 375015501*x^10/10! + 14973327740*x^11/11! + 693507063942*x^12/12! +...
such that A( [Integral A(x) dx] / A(x) ) = 1 + x.
RELATED SERIES.
[Integral A(x) dx] / A(x) = x - x^2/2! + x^3/3! - 4*x^4/4! - 6*x^5/5! - 189*x^6/6! - 2870*x^7/7! - 66500*x^8/8! - 1828400*x^9/9! - 60761407*x^10/10! - 2374495340*x^11/11! - 107581374684*x^12/12! +...
which equals Series_Reversion( A(x) - 1 ).
PROG
(PARI) {a(n) = my(A=[1, 1], F); for(i=1, n+1, A = concat(A, 0); F=sum(m=1, #A, A[m]*x^(m-1)/(m-1)!) +x*O(x^#A); A[#A] = -(#A-1)!*polcoeff( subst(F, x, intformal(F)/F ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 24 2016
STATUS
approved