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A201465 E.g.f. satisfies: A(x) = (x + 2*exp(A(x)) - 2)/3.

%I A201465 #27 Mar 21 2025 18:28:39
%S A201465 1,2,14,162,2622,54546,1386702,41660226,1444071006,56728401138,
%T A201465 2490626473326,120858220146978,6423145784929278,371046277074303954,
%U A201465 23148851187463826958,1551182540888019542274,111111330526583477368734,8472364399282482984295602,685178683361064789536947374
%N A201465 E.g.f. satisfies: A(x) = (x + 2*exp(A(x)) - 2)/3.
%F A201465 E.g.f. A(x) satisfies: x = A( 2 + 3*x - 2*exp(x) ).
%F A201465 a(n)=(sum(k=0..n-1, (n+k-1)!*sum(j=0..k, 1/(k-j)!*sum(i=0..j, (3^i*(-1)^(i)*2^(j-i)*stirling2(n+j-i-1,j-i))/(i!*(n+j-i-1)!))))), n>0. [From Vladimir Kruchinin, Feb 04 2012]
%F A201465 exp(A(x))-1 is the compositional inverse of 3*log(1+x)-2*x and is the e.g.f. of A058562. - _Peter Bala_, Jul 12 2012
%F A201465 G.f.: 1/Q(0), where Q(k)= 1 - k*x - 2*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 01 2013
%F A201465 E.g.f.: (x-2)/3 - LambertW(-2/3*exp((x-2)/3)). - _Vaclav Kotesovec_, Dec 26 2013
%F A201465 a(n) ~ n^(n-1) / (sqrt(3) * exp(n) * (3*log(3)-3*log(2)-1)^(n-1/2)). - _Vaclav Kotesovec_, Dec 26 2013
%F A201465 O.g.f.: Sum_{n>=0} 2^n / Product_{k=0..n} (3 - k*x). - _Paul D. Hanna_, Oct 27 2014
%e A201465 E.g.f.: A(x) = x + 2*x^2 + 14*x^3/3! + 162*x^4/4! + 2622*x^5/5! + 54546*x^6/6! +...
%e A201465 The exponential of the e.g.f. begins:
%e A201465 exp(A(x)) = 1 + x + 3*x^2/2! + 21*x^3/3! + 243*x^4/4! + 3933*x^5/5! + 81819*x^6/6! +...
%e A201465 where x = 2 + 3*A(x) - 2*exp(A(x)).
%e A201465 ...
%e A201465 O.g.f.: G(x) = 1 + 2*x + 14*x^2 + 162*x^3 + 2622*x^4 + 54546*x^5 +...
%e A201465 where
%e A201465 G(x) = 1/3 + 2/(3*(3-x)) + 2^2/(3*(3-x)*(3-2*x)) + 2^3/(3*(3-x)*(3-2*x)*(3-3*x)) + 2^4/(3*(3-x)*(3-2*x)*(3-3*x)*(3-4*x)) + 2^5/(3*(3-x)*(3-2*x)*(3-3*x)*(3-4*x)*(3-5*x)) +...
%t A201465 Rest[CoefficientList[1 + InverseSeries[Series[2 + 3*x - 2*Exp[x], {x, 0, 20}], x],x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Dec 26 2013 *)
%o A201465 (PARI) {a(n)=n!*polcoeff(serreverse(2+3*x - 2*exp(x+x^2*O(x^n))),n)}
%o A201465 for(n=0, 25, print1(round(A[n+1]), ", "))
%o A201465 (PARI) \p100 \\ set precision
%o A201465 {A=Vec(sum(n=0, 600, 1.*2^n/prod(k=0, n, 3 - k*x + O(x^31))))}
%o A201465 for(n=0, 25, print1(round(A[n+1]), ", ")) \\ _Paul D. Hanna_, Oct 27 2014
%o A201465 (Maxima) a(n):=(sum((n+k-1)!*sum(1/(k-j)!*sum((3^i*(-1)^(i)*2^(j-i)*stirling2(n+j-i-1,j-i))/(i!*(n+j-i-1)!),i,0,j),j,0,k),k,0,n-1)); /* Vladimir Kruchinin, Feb 04 2012 */
%Y A201465 Cf. variants: A000311, A201466. A058562.
%K A201465 nonn
%O A201465 1,2
%A A201465 _Paul D. Hanna_, Dec 01 2011