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A277184
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E.g.f.: A(x) = x*exp(A(x) - A(x)^2) + A(x)^2.
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2
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1, 4, 33, 424, 7505, 170496, 4744873, 156529024, 5974216641, 258970009600, 12566664261041, 674795685758976, 39720422453156497, 2543022838953017344, 175923061842374645625, 13076498369827187163136, 1039320236257785348449537, 87954586779787961844105216, 7895887532418683295505005121, 749448035808323155802521600000, 74989090946223628553344278643281, 7888932153987131087072869161631744
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OFFSET
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1,2
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LINKS
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FORMULA
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E.g.f. A(x) satisfies:
(1) exp(A(x) - A(x)^2) = LambertW(-x)/(-x).
(2) A(x) = -LambertW(-x) + A(x)^2.
(3) A(x) = C( -LambertW(-x) ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers.
a(n) = Sum_{k=1..n} A000108(k-1) * n^(n-k) * k! * binomial(n-1,k-1), where A000108 is the Catalan numbers.
a(n) ~ 2^(2*n - 1/2) * n^(n-1) / (sqrt(3) * exp(3*n/4)). - Vaclav Kotesovec, Oct 10 2016
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EXAMPLE
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E.g.f.: A(x) = x + 4*x^2/2! + 33*x^3/3! + 424*x^4/4! + 7505*x^5/5! + 170496*x^6/6! + 4744873*x^7/7! + 156529024*x^8/8! + 5974216641*x^9/9! + 258970009600*x^10/10! +...
such that
A(x) - A(x)^2 = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! + 7776*x^6/6! + 117649*x^7/7! +...+ n^(n-1)*x^n/n! +...
which equals -LambertW(-x).
RELATED SERIES.
A(x)^2 = 2*x^2/2! + 24*x^3/3! + 360*x^4/4! + 6880*x^5/5! + 162720*x^6/6! + 4627224*x^7/7! + 154431872*x^8/8! + 5931169920*x^9/9! + 257970009600*x^10/10! +...
exp(A(x)) = 1 + x + 5*x^2/2! + 46*x^3/3! + 629*x^4/4! + 11556*x^5/5! + 268537*x^6/6! + 7578040*x^7/7! + 252168009*x^8/8! + 9677553040*x^9/9! + 421010089901*x^10/10! +...
exp(A(x)^2) = 1 + 2*x^2/2! + 24*x^3/3! + 372*x^4/4! + 7360*x^5/5! + 179400*x^6/6! + 5228664*x^7/7! + 177953552*x^8/8! + 6940738368*x^9/9! + 305570622240*x^10/10! +...
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MATHEMATICA
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Rest[CoefficientList[Series[(1 - Sqrt[1 + 4*LambertW[-x]])/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Oct 10 2016 *)
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PROG
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(PARI) {a(n) = sum(k=1, n, n^(n-k) * (2*k-2)!/(k-1)!^2 * (n-1)!/(n-k)! )}
for(n=1, 25, print1(a(n), ", "))
(PARI) {a(n) = my(A=x); for(i=0, n, A = x*exp(A - A^2 +x*O(x^n)) + A^2 ); n!*polcoeff(A, n)}
for(n=1, 25, 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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