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A268654 E.g.f.: exp( T(T(T(T(x)))) ), where T(x) = -LambertW(-x) is Euler's tree function (A000169). 2
1, 1, 9, 133, 2729, 71721, 2300485, 87194689, 3815719969, 189440927857, 10525328121221, 647265172064985, 43660242639018241, 3205987437435132793, 254635755560090281525, 21755037223870035810001, 1989746853200670755116865, 194000891136578173746676449, 20089033883934411591428091013, 2202022786357483714102765694185 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

FORMULA

E.g.f. satisfies:

(1) A(x) = A(x/exp(x))^A(x).

(2) A(x) = exp( A(x)*T(T(T(x))) ).

(3) A(x/exp(x)) = exp(T(T(T(x)))) = LambertW(LambertW(LambertW(-x))) / LambertW(LambertW(-x)).

a(n) ~ exp(1 + (exp(-1) + exp(-1 - exp(-1)) + exp(-1 - exp(-1) - exp(-1 - exp(-1))))*n) * n^(n-1) / sqrt((1 - exp(-1)) * (1 + LambertW(LambertW(-exp(-1 - exp(-1) - exp(-1 - exp(-1)) - exp(-1 - exp(-1) - exp(-1 - exp(-1))))))) * (1 + LambertW(-exp(-1 - exp(-1) - exp(-1 - exp(-1)) - exp(-1 - exp(-1) - exp(-1 - exp(-1))))))). - Vaclav Kotesovec, Apr 01 2016

EXAMPLE

E.g.f.: A(x) = 1 + x + 9*x^2/2! + 133*x^3/3! + 2729*x^4/4! + 71721*x^5/5! + 2300485*x^6/6! + 87194689*x^7/7! + 3815719969*x^8/8! +...

where A(x) = A( x/exp(x) )^A(x).

RELATED SERIES.

Define W(x) = LambertW(-x)/(-x), where W(x) = exp(x*W(x)) and begins:

W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + 7^5*x^6/6! + 8^6*x^7/7! + 9^7*x^8/8! +...+ A000272(n+1)*x^n/n! +...

Let F(x) = A(x/exp(x)), which begins:

F(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 525*x^4/4! + 8321*x^5/5! + 162463*x^6/6! + 3774513*x^7/7! + 101808185*x^8/8! +...+ A227176(n)*x^n/n! +...

Let G(x) = F(x/exp(x)), which begins:

G(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! +...

then W(x), F(x), G(x), and A(x) are in the family of functions that begin:

(1) W(x) = exp(x)^W(x) = exp(T(x)),

(2) F(x) = W(x)^F(x) = exp(T(T(x))),

(3) G(x) = F(x)^G(x) = exp(T(T(T(x)))),

(4) A(x) = G(x)^A(x) = exp(T(T(T(T(x))))), ...

where T(x) = -LambertW(-x) is Euler's tree function:

T(x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + 6^5*x^6/! + 7^6*x^7/7! + 8^7*x^8/8! +...+ A000169(n)*x^n/n! +...

MATHEMATICA

CoefficientList[Series[E^(-ProductLog[ProductLog[ProductLog[ProductLog[-x]]]]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Apr 01 2016 *)

PROG

(PARI) /* E.g.f.: A(x) = exp(T(T(T(T(x)))) ) */

{a(n)=local(T=sum(k=1, n, k^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(exp(subst(T, x, subst(T, x, subst(T, x, T)))), n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) /* E.g.f.: A(x) = exp( -A(x)*LambertW(LambertW(LambertW(-x))) ) */

{a(n)=local(A=1+x, LambertW=sum(k=1, n, -k^(k-1)*(-x)^k/k!)+x*O(x^n));

for(i=1, n, A=exp(-A*subst(LambertW, x, subst(LambertW, x, subst(LambertW, x, -x))) +x*O(x^n))); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A268653, A227176, A000272, A000169.

Sequence in context: A089547 A110273 A082760 * A112426 A213688 A163200

Adjacent sequences:  A268651 A268652 A268653 * A268655 A268656 A268657

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 18 2016

STATUS

approved

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Last modified January 24 07:18 EST 2020. Contains 331189 sequences. (Running on oeis4.)