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A203038
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E.g.f.: 1 - (1-x)^(1/(1 + log(1-x))).
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1
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1, 2, 6, 22, 80, 98, -4214, -99780, -1845204, -32904168, -595817376, -11140397424, -216067092696, -4341986018424, -89974228233816, -1906379451494736, -40730527391178288, -855960386735708448, -16775306003344482336, -260505777142558363488, -292631259982475443392
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OFFSET
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1,2
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COMMENTS
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E.g.f. A(x) = (1 - exp(-x)) o x/(1-x) o -log(1-x), a composition of functions such that the n-th iteration of A(x) equals (1 - exp(-x)) o x/(1-n*x) o -log(1-x).
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LINKS
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Table of n, a(n) for n=1..21.
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EXAMPLE
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E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 22*x^4/4! + 80*x^5/5! + 98*x^6/6! +...
Illustration of the initial iterations of A(x).
The 2nd iteration of A(x) equals:
A(A(x)) = 1 - (1-x)^(1/(1 + 2*log(1-x))), which begins:
A(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 188*x^4/4! + 1780*x^5/5! +...
The 3rd iteration of A(x) equals:
A(A(A(x))) = 1 - (1-x)^(1/(1 + 3*log(1-x))), which begins:
A(A(A(x))) = x + 6*x^2/2! + 54*x^3/3! + 642*x^4/4! + 9420*x^5/5! +...
The 4th iteration of A(x) equals:
A(A(A(A(x)))) = 1 - (1-x)^(1/(1 + 4*log(1-x))), which begins:
A(A(A(A(x)))) = x + 8*x^2/2! + 96*x^3/3! + 1528*x^4/4! + 30200*x^5/5! +...
etc.
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MATHEMATICA
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With[{nn=30}, Rest[CoefficientList[Series[1-(1-x)^(1/(1+Log[1-x])), {x, 0, nn}], x]Range[0, nn]!]] (* Harvey P. Dale, Sep 22 2012 *)
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PROG
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(PARI) /* a(n, k) = the coefficient of x^n/n! in the k-th iteration: */
{a(n, k=1)=n!*polcoeff(1-(1-x+x*O(x^n))^(1/(1+k*log(1-x+x*O(x^n)))), n)}
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CROSSREFS
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Cf. A203039, A000629.
Sequence in context: A181367 A106434 A150228 * A206304 A201372 A072547
Adjacent sequences: A203035 A203036 A203037 * A203039 A203040 A203041
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KEYWORD
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sign
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AUTHOR
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Paul D. Hanna, Dec 27 2011
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STATUS
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approved
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