A276231 E.g.f. A(x) satisfies: A(x)^A(x) = LambertW(-x)/(-x).

1, 1, 1, 7, 37, 441, 4771, 79213, 1320649, 28318321, 636978151, 16863972621, 475580960317, 15055752973561, 508984025190187, 18802677669334861, 739723172361876241, 31282037176343362785, 1402437758091393319759, 66859536126516402568717, 3362832363918613596662341, 178500985406930615357763241, 9950984335825184802962609491, 582129154096893229447821411597, 35620632904151979409688095495897, 2277073896917989779381561818509201

Offset: 0

Author

Paul D. Hanna, Aug 24 2016

Keywords

• nonn

Comments

Let G(x) = (-x)/LambertW(-x), then A(x)^A(x) = 1/G(x) where G(x)^G(x) = 1/exp(x).

a(n) = 0 (mod 3) when n = 6*k+5, k>=0, otherwise a(n) = 1 (mod 3).

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 7*x^3/3! + 37*x^4/4! + 441*x^5/5! + 4771*x^6/6! + 79213*x^7/7! + 1320649*x^8/8! + 28318321*x^9/9! + 636978151*x^10/10! + 16863972621*x^11/11! + 475580960317*x^12/12! +...
such that A(x)^A(x) = LambertW(-x)/(-x), where
LambertW(-x)/(-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! +...+ (n+1)^(n-1)*x^n/n! +...
The logarithm of the e.g.f. A(x) begins
log(A(x)) = x + 6*x^3/3! + 12*x^4/4! + 320*x^5/5! + 2190*x^6/6! + 51492*x^7/7! + 685496*x^8/8! + 17286768*x^9 +...+ A097174(n)*x^n/n! +...
which equals -T(-T(x)), where
T(x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! + 7776*x^6/6! + 117649*x^7/7! + 2097152*x^8/8! +...+ n^(n-1)*x^n/n! +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A097174 (log(A(x))).

Programs

Mathematica

CoefficientList[Series[E^LambertW[-LambertW[-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;
for(i=0,n, A = W^(1/A) ); n!*polcoeff(A,n)}
for(n=0, 40, print1(a(n), ", "))

Formula

E.g.f.: exp( -T(-T(x)) ), where T(x) = Sum_{n>=1} n^(n-1)*x^n/n!.

a(n) ~ n^(n-1)/(1+LambertW(1)). - Vaclav Kotesovec, Aug 26 2016