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A276742
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E.g.f. A(x) satisfies: A( 1/A(x) * Integral A(x) dx ) = 1 + x.
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0
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1, 1, 1, 2, 9, 76, 980, 17304, 393463, 11072376, 375015501, 14973327740, 693507063942, 36782159095080, 2210369895001450, 149163550608705780, 11218246110724502325, 934089674706365890832, 85613718583699681233208, 8593417105404547807210554, 940306481313403267058365853, 111703159299047925885976523740, 14352698749278209896668891217608, 1987913848425789150258188910598408, 295873533681557805541331625248339120
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OFFSET
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0,4
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LINKS
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FORMULA
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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.
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EXAMPLE
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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 ).
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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