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%I #6 Sep 29 2016 19:50:16
%S 1,1,1,2,9,76,980,17304,393463,11072376,375015501,14973327740,
%T 693507063942,36782159095080,2210369895001450,149163550608705780,
%U 11218246110724502325,934089674706365890832,85613718583699681233208,8593417105404547807210554,940306481313403267058365853,111703159299047925885976523740,14352698749278209896668891217608,1987913848425789150258188910598408,295873533681557805541331625248339120
%N E.g.f. A(x) satisfies: A( 1/A(x) * Integral A(x) dx ) = 1 + x.
%F E.g.f. A(x) satisfies:
%F (1) A(x) = 1 + Series_Reversion( 1/A(x) * Integral A(x) dx ).
%F (2) Integral A(x) dx = A(x) * Series_Reversion( A(x) - 1 ).
%F (3) A(x) = exp( Integral (1 - B'(x))/B(x) dx ), where B(A(x) - 1) = x.
%e 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! +...
%e such that A( [Integral A(x) dx] / A(x) ) = 1 + x.
%e RELATED SERIES.
%e [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! +...
%e which equals Series_Reversion( A(x) - 1 ).
%o (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]}
%o for(n=0,30,print1(a(n),", "))
%K nonn
%O 0,4
%A _Paul D. Hanna_, Sep 24 2016
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