A201466 E.g.f. satisfies: A(x) = (x + 3*exp(A(x)) - 3)/4.
1, 3, 30, 498, 11568, 345432, 12606240, 543678672, 27054328512, 1525746223488, 96167433279360, 6699404849841408, 511152613463843328, 42391161255859802112, 3796840836492517125120, 365260399012767192102912, 37561729737177160757133312, 4111876748834828077514170368
Offset: 1
Keywords
Examples
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! +... The exponential of the e.g.f. begins: 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! +... where x = 3 + 4*A(x) - 3*exp(A(x)). ... O.g.f.: G(x) = 1 + 3*x + 30*x^2 + 498*x^3 + 11568*x^4 + 345432*x^5 +... where 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)) +...
Programs
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Mathematica
Rest[CoefficientList[InverseSeries[Series[3 - 3*E^x + 4*x,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *) -
PARI
{a(n)=n!*polcoeff(serreverse(3+4*x - 3*exp(x+x^2*O(x^n))),n)} -
PARI
\p100 \\ set precision {A=Vec(sum(n=0, 600, 1.*3^n/prod(k=0, n, 4 - k*x + O(x^31))))} for(n=0, 25, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 27 2014
Formula
E.g.f. satisfies: x = A( 3 + 4*x - 3*exp(x) ).
E.g.f.: (x-3)/4 - LambertW(-3*exp((x-3)/4)/4). - Vaclav Kotesovec, Jan 10 2014
a(n) ~ n^(n-1) / (2 * (4*log(4/3)-1)^(n-1/2) * exp(n)). - Vaclav Kotesovec, Jan 10 2014
O.g.f.: Sum_{n>=0} 3^n / Product_{k=0..n} (4 - k*x). - Paul D. Hanna, Oct 27 2014