A276232 E.g.f. W(x)^W(x), where W(x) = LambertW(-x)/(-x) and satisfies W(x)^(1/W(x)) = exp(x).

1, 1, 5, 37, 369, 4641, 70513, 1256361, 25689569, 592998049, 15254145441, 432741923769, 13422771397489, 451956633181041, 16418360947494353, 640101032269166281, 26659869619363530177, 1181426235219348415041, 55506506359841719631425, 2756055067645593490566489, 144211201693287134720633681, 7931567651481815767204102801

Offset: 0

Author

Paul D. Hanna, Aug 24 2016

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 369*x^4/4! + 4641*x^5/5! + 70513*x^6/6! + 1256361*x^7/7! + 25689569*x^8/8! + 592998049*x^9/9! + 15254145441*x^10/10! +...
such that A(x) = W(x)^W(x).
Define W(x) = LambertW(-x)/(-x), where
W(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! + 100000000*x^9/9! +...+ (n+1)^(n-1)*x^n/n! +...
satisfies W(x) = exp(x)^W(x).
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..384

Crossrefs

Cf. A276231.

Programs

Mathematica

CoefficientList[Series[E^(LambertW[-x]^2/x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 26 2016 *)

PARI

{a(n) = my(A=1+x, W); W=serreverse(x*exp(-x +x^2*O(x^n)))/x; A = W^W; n!*polcoeff(A,n)}
for(n=0, 40, print1(a(n), ", "))

Formula

E.g.f.: exp( LambertW(-x)^2/x ).

a(n) ~ 2*exp(exp(1)+1)*n^(n-1). - Vaclav Kotesovec, Aug 26 2016