login
E.g.f. A(x) satisfies: A(x)^A(x) = LambertW(-x)/(-x).
3

%I #28 Aug 26 2016 02:55:53

%S 1,1,1,7,37,441,4771,79213,1320649,28318321,636978151,16863972621,

%T 475580960317,15055752973561,508984025190187,18802677669334861,

%U 739723172361876241,31282037176343362785,1402437758091393319759,66859536126516402568717,3362832363918613596662341,178500985406930615357763241,9950984335825184802962609491,582129154096893229447821411597,35620632904151979409688095495897,2277073896917989779381561818509201

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

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

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

%H Paul D. Hanna, <a href="/A276231/b276231.txt">Table of n, a(n) for n = 0..200</a>

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

%F a(n) ~ n^(n-1)/(1+LambertW(1)). - _Vaclav Kotesovec_, Aug 26 2016

%e 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! +...

%e such that A(x)^A(x) = LambertW(-x)/(-x), where

%e 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! +...

%e The logarithm of the e.g.f. A(x) begins

%e 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! +...

%e which equals -T(-T(x)), where

%e 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! +...

%t CoefficientList[Series[E^LambertW[-LambertW[-x]], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Aug 26 2016 *)

%o (PARI) {a(n) = my(A=1+x, W); W=serreverse(x*exp(-x +x^2*O(x^n)))/x;

%o for(i=0,n, A = W^(1/A) ); n!*polcoeff(A,n)}

%o for(n=0, 40, print1(a(n), ", "))

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

%K nonn

%O 0,4

%A _Paul D. Hanna_, Aug 24 2016