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A274275
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E.g.f. A(x) satisfies: A( sqrt( A(x^2*exp(-2*x)) ) ) = x.
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6
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1, 2, 6, 40, 400, 4656, 62944, 1046144, 20274048, 438238720, 10529132416, 280439144448, 8185848206848, 259202608222208, 8855252721592320, 324989707586830336, 12748521382531956736, 532098814401540587520, 23547710868033300004864, 1101540715832518509854720, 54307901369414002580422656, 2814303585179538846791237632, 152935335939643406489642008576, 8696644113583584719506275041280, 516469893784923819203984490496000
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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) = Sum_{n>=1} a(n) * x^n / n! satisfies:
(1) A( sqrt( A(x^2*exp(2*x)) ) ) = -LambertW(-x*exp(x)).
(2) A(x) = Series_Reversion( sqrt( A(x^2*exp(-2*x)) ) ).
(3) A( A(x)^2 * exp(-2*A(x)) ) = x^2.
(4) A(-A(x)^2 * exp(-2*A(x)) ) = -LambertW(x^2*exp(-x^2)).
a(n)/n! ~ c * d^n / n^(3/2), where d = 2.52462188117..., c = 0.36965356... . - Vaclav Kotesovec, Jun 24 2016
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EXAMPLE
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E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 40*x^4/4! + 400*x^5/5! + 4656*x^6/6! + 62944*x^7/7! + 1046144*x^8/8! + 20274048*x^9/9! + 438238720*x^10/10! + 10529132416*x^11/11! + 280439144448*x^12/12! + 8185848206848*x^13/13! + 259202608222208*x^14/14! + 8855252721592320*x^15/15! + 324989707586830336*x^16/16! +...
such that A( sqrt( A(x^2*exp(-2*x)) ) ) = x.
RELATED SERIES.
The series reversion of the e.g.f. A(x) equals the series defined by:
sqrt( A(x^2*exp(-2*x)) ) = x - 2*x^2/2! + 6*x^3/3! - 40*x^4/4! + 320*x^5/5! - 2976*x^6/6! + 35392*x^7/7! - 538112*x^8/8! + 9931392*x^9/9! - 211790080*x^10/10! + 5059784576*x^11/11! - 132643057152*x^12/12! + 3761875287040*x^13/13! - 114501941915648*x^14/14! + 3725395402721280*x^15/15! - 129324055589257216*x^16/16! +...+ A274277(n)*x^n/n! +...
Compare the above series reversion to the following series:
A(x)^2 * exp(-2*A(x)) = x^2 - 2*x^4/2! + 6*x^6/3! - 40*x^8/4! + 320*x^10/5! - 2976*x^12/6! + 35392*x^14/7! - 538112*x^16/8! + 9931392*x^18/9! +...
where A( A(x)^2 * exp(-2*A(x)) ) = x^2.
The e.g.f. A(x) is related to the LambertW function by the composition:
A( sqrt( A(x^2*exp(2*x)) ) ) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 47232*x^6/6! + 942592*x^7/7! + 22171648*x^8/8! +...+ A216857(n)*x^n/n! +...
which equals -LambertW(-x*exp(x)).
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PROG
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(PARI) {a(n) = my(A=x); for(i=1, n, A = serreverse( sqrt( subst(A, x, x^2*exp(-2*x +x*O(x^n))) ) ) ); n!*polcoeff(A, n)}
for(n=1, 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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