%I #11 Oct 27 2014 22:59:55

%S 1,3,30,498,11568,345432,12606240,543678672,27054328512,1525746223488,

%T 96167433279360,6699404849841408,511152613463843328,

%U 42391161255859802112,3796840836492517125120,365260399012767192102912,37561729737177160757133312,4111876748834828077514170368

%N E.g.f. satisfies: A(x) = (x + 3*exp(A(x)) - 3)/4.

%F E.g.f. satisfies: x = A( 3 + 4*x - 3*exp(x) ).

%F E.g.f.: (x-3)/4 - LambertW(-3*exp((x-3)/4)/4). - _Vaclav Kotesovec_, Jan 10 2014

%F a(n) ~ n^(n-1) / (2 * (4*log(4/3)-1)^(n-1/2) * exp(n)). - _Vaclav Kotesovec_, Jan 10 2014

%F O.g.f.: Sum_{n>=0} 3^n / Product_{k=0..n} (4 - k*x). - _Paul D. Hanna_, Oct 27 2014

%e E.g.f.: A(x) = x + 3*x^2/2! + 30*x^3/3! + 498*x^4/4! + 11568*x^5/5! + 345432*x^6/6! +...

%e The exponential of the e.g.f. begins:

%e exp(A(x)) = 1 + x + 4*x^2/2! + 40*x^3/3! + 664*x^4/4! + 15424*x^5/5! + 460576*x^6/6! +...

%e where x = 3 + 4*A(x) - 3*exp(A(x)).

%e ...

%e O.g.f.: G(x) = 1 + 3*x + 30*x^2 + 498*x^3 + 11568*x^4 + 345432*x^5 +...

%e where

%e G(x) = 1/4 + 3/(4*(4-x)) + 3^2/(4*(4-x)*(4-2*x)) + 3^3/(4*(4-x)*(4-2*x)*(4-3*x)) + 3^4/(4*(4-x)*(4-2*x)*(4-3*x)*(4-4*x)) + 3^5/(4*(4-x)*(4-2*x)*(4-3*x)*(4-4*x)*(4-5*x)) +...

%t Rest[CoefficientList[InverseSeries[Series[3 - 3*E^x + 4*x,{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 10 2014 *)

%o (PARI) {a(n)=n!*polcoeff(serreverse(3+4*x - 3*exp(x+x^2*O(x^n))),n)}

%o (PARI) \p100 \\ set precision

%o {A=Vec(sum(n=0, 600, 1.*3^n/prod(k=0, n, 4 - k*x + O(x^31))))}

%o for(n=0, 25, print1(round(A[n+1]), ", ")) \\ _Paul D. Hanna_, Oct 27 2014

%Y Cf. variants: A000311, A201465.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Dec 01 2011