A268653 E.g.f.: exp( T(T(T(x))) ), where T(x) = -LambertW(-x) is Euler's tree function (A000169).

1, 1, 7, 82, 1345, 28396, 734149, 22485898, 796769201, 32084546824, 1447917011461, 72411962077126, 3976481464087609, 237939307837951708, 15412492927027232261, 1074675869343994244266, 80270802348342665849569, 6395153963612453962942096, 541390375948749181692141061, 48536543026953818449535683054, 4594206854845500504888845269481, 457878082780635055560866092165156, 47930551834845432770784732668907205

Offset: 0

Author

Paul D. Hanna, Feb 09 2016

Keywords

• nonn

Examples

			E.g.f.: 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! +...
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! +...
then
(1) A(x) = W( x*W(x) * W(x*W(x)) ),
(2) A(x) = W( x*W(x) )^A(x),
(3) A(x) = exp( A(x) * x*W(x) * W(x*W(x)) ),
(4) A(x/exp(x)) = W(x*W(x)).
Let G(x) = A(x/exp(x)), which begins:
G(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! +...
then W(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) G(x) = W(x)^G(x) = exp(T(T(x))),
(3) A(x) = G(x)^A(x) = exp(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! +...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A227176, A227278, A000169, A000272.

Programs

PARI

/* E.g.f.: A(x) = exp(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, T))), n)}
for(n=0, 25, print1(a(n), ", "))

PARI

/* E.g.f.: A(x) = W( x*W(x) * W(x*W(x)) ) */
{a(n)=local(W=sum(k=0, n, (k+1)^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(subst(W, x, subst(x*W, x, x*W)), n)}
for(n=0, 25, print1(a(n), ", "))

PARI

/* E.g.f.: A(x) = exp( -A(x)*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, -x)) +x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

PARI

/* E.g.f.: A(x) = ( LambertW(LambertW(-x))/LambertW(-x) )^A(x) */
{a(n)=local(A=1+x, W=sum(k=0, n, (k+1)^(k-1)*x^k/k!)+x*O(x^n));
for(i=1, n, A=subst(W,x,x*W)^A); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f. satisfies:

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

(2) A(x) = W( x*W(x) * W(x*W(x)) ), where W(x) = LambertW(-x)/(-x).

(3) A(x) = W( x*W(x) )^A(x), where W(x) = LambertW(-x)/(-x).

(4) A(x) = exp( -A(x)*LambertW(LambertW(-x)) ).

(5) A(x) = ( LambertW(LambertW(-x)) / LambertW(-x) )^A(x).

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

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