The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A227278 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A227278

E.g.f.: T(T(T(x))), where T(x) = -LambertW(-x) is Euler's tree function (A000169).
(history; published version)

#14 by Paul D. Hanna at Thu Feb 11 12:37:50 EST 2016

STATUS

editing

approved

#13 by Paul D. Hanna at Thu Feb 11 12:37:44 EST 2016

EXAMPLE

A(x/exp(x)) = A(x)/exp(A(x)) = x + 4*x^2/2! + 30*x^3/3! + 332*x^4/4! + 4880*x^5/5! + 89742*x^6/6! + 1986124*x^7/7! + 51471800*x^8/8! +...+ A207833(n)*x^n/n! +...

exp(A(x)) = 1 + x + 7*x^2/2! + 82*x^3/3! + 1345*x^4/4! + 28396*x^5/5! + 734149*x^6/6! + 22485898*x^7/7! + 796769201*x^8/8! +...+ A268653(n)*x^n/n! +...

CROSSREFS

Cf. A268653, A207833, A000169.

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approved

editing

#12 by Alois P. Heinz at Fri Jul 05 10:34:21 EDT 2013

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approved

#11 by Alois P. Heinz at Fri Jul 05 10:34:16 EDT 2013

LINKS

Alois P. Heinz, <a href="/A227278/b227278.txt">Table of n, a(n) for n = 1..100</a>

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proposed

editing

#10 by Vaclav Kotesovec at Fri Jul 05 09:53:10 EDT 2013

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editing

proposed

#9 by Vaclav Kotesovec at Fri Jul 05 09:50:37 EDT 2013

MATHEMATICA

Rest[CoefficientList[Series[-LambertW[LambertW[LambertW[-x]]], {x, 0, 20}], x]* Range[0, 20]! ] (* Vaclav Kotesovec, Jul 05 2013 *)

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approved

editing

Discussion

Fri Jul 05

09:51

Vaclav Kotesovec: Done.

09:53

Vaclav Kotesovec: (code corrected for offset 1)

#8 by Paul D. Hanna at Fri Jul 05 09:20:02 EDT 2013

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proposed

approved

#7 by Vaclav Kotesovec at Fri Jul 05 05:52:50 EDT 2013

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editing

proposed

Discussion

Fri Jul 05

09:19

Paul D. Hanna: Quite impressive asymptotics, Vaclav!  
I've tested your formula, and it is good.  Thank you!

09:29

Wesley Ivan Hurt: Can you change the Mathematica code to reflect the offset?

#6 by Vaclav Kotesovec at Fri Jul 05 05:52:41 EDT 2013

FORMULA

a(n) ~ n! * exp((1+exp(-1)+exp(-1-exp(-1)))*n)/(sqrt(2*Pi*(1-exp(-1))*(1-exp(-1-exp(-1))))*n^(3/2)). - Vaclav Kotesovec, Jul 05 2013

MATHEMATICA

CoefficientList[Series[-LambertW[LambertW[LambertW[-x]]], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 05 2013 *)

STATUS

approved

editing

#5 by Paul D. Hanna at Thu Jul 04 21:33:43 EDT 2013

STATUS

editing

approved