A227278 E.g.f.: T(T(T(x))), where T(x) = -LambertW(-x) is Euler's tree function (A000169).

1, 6, 63, 948, 18645, 454158, 13221075, 448434136, 17386204761, 759123121050, 36882981687519, 1974616464026484, 115536647641839333, 7336947898087080406, 502660682907018997755, 36961205206337621142192, 2903732354672613314658225, 242753209611983811853905330

Offset: 1

Author

Paul D. Hanna, Jul 04 2013

Keywords

• nonn

Examples

			E.g.f.: A(x) = x + 6*x^2/2! + 63*x^3/3! + 948*x^4/4! + 18645*x^5/5! +...
Euler's tree function T(x) satisfies: T(x/exp(x)) = x, and begins:
T(x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! +...
where A(x) = T(T(T(x))).
Related expansions:
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! +...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A268653, A207833, A000169.

Programs

Mathematica

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

PARI

/* E.g.f.: A(x) = T(T(T(x))) */
{a(n)=local(T=sum(k=1, n, k^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(subst(T,x,subst(T, x, T)), n)}
for(n=1, 20, print1(a(n), ", "))

PARI

/* E.g.f.: A(x) = -LambertW(LambertW(LambertW(-x))) */
{a(n)=local(LambertW=sum(k=1, n, -k^(k-1)*(-x)^k/k!)+x*O(x^n));
n!*polcoeff(-subst(LambertW,x,subst(LambertW,x,subst(LambertW,x,-x))),n)}
for(n=1, 20, print1(a(n), ", "))

Formula

Given e.g.f. A(x), A(x/exp(x)) = A(x)/exp(A(x)) = T(T(x)) and equals the e.g.f. of A207833.

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