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A192949
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E.g.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^2 - 1)^n/n!.
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1
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1, 1, 4, 42, 704, 16300, 482112, 17366776, 737738752, 36109329552, 2001104000000, 123856655495584, 8468525621182464, 633915692700252352, 51562270240172425216, 4528439794201950000000, 427082984690083973562368, 43049504748861000404766976
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f. A(x) equals the formal inverse of function (x-1)/exp(x^2-1).
E.g.f. satisfies: A(x) = 1 + x*exp(A(x)^2-1).
E.g.f.: A(x) = 1 + Series_Reversion( x/exp(2*x + x^2) ).
E.g.f. satisfies: A(x/G(x)) = 1 + x where G(x) = exp(2*x+x^2) = x/Series_Reversion(A(x)-1) = e.g.f. of A000898.
a(n) ~ (1+sqrt(3))^n * exp((sqrt(3)/2-1)*n) * n^(n-1)/sqrt(2*(3+sqrt(3))). - Vaclav Kotesovec, Feb 26 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 4*x^2/2! + 42*x^3/3! + 704*x^4/4! + 16300*x^5/5! +...
where (A(x) - 1)/exp(A(x)^2-1) = x.
Related expansions.
(A(x)^2-1) = 2*x + 10*x^2/2! + 108*x^3/3! + 1840*x^4/4! + 43000*x^5/5! +...
(A(x)^2-1)^2 = 8*x^2/2! + 120*x^3/3! + 2328*x^4/4! + 58400*x^5/5! +...
(A(x)^2-1)^3 = 48*x^3/3! + 1440*x^4/4! + 43920*x^5/5! +...
(A(x)^2-1)^4 = 384*x^4/4! + 19200*x^5/5! + 846720*x^6/6! +...
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MATHEMATICA
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CoefficientList[1 + InverseSeries[Series[x/E^(2*x + x^2), {x, 0, 20}], x], x]*Range[0, 20]! (* Vaclav Kotesovec, Feb 26 2014 *)
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PROG
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(PARI) {a(n)=local(A=1+serreverse(x/exp(2*x+x^2+x^2*O(x^n)))); n!*polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*exp(A^2-1+x*O(x^n))); n!*polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*sum(m=0, n, (A^2-1+x*O(x^n))^m/m!)); n!*polcoeff(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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