A268654 E.g.f.: exp( T(T(T(T(x)))) ), where T(x) = -LambertW(-x) is Euler's tree function (A000169).

1, 1, 9, 133, 2729, 71721, 2300485, 87194689, 3815719969, 189440927857, 10525328121221, 647265172064985, 43660242639018241, 3205987437435132793, 254635755560090281525, 21755037223870035810001, 1989746853200670755116865, 194000891136578173746676449, 20089033883934411591428091013, 2202022786357483714102765694185

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