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A136727 E.g.f.: A(x) = (exp(x)/(3 - 2*exp(x)))^(1/3).

%I A136727 #32 May 06 2025 06:52:46
%S A136727 1,1,3,17,139,1481,19443,303297,5480219,112549881,2589274883,
%T A136727 65957355377,1842897053099,56038776055081,1842278768795923,
%U A136727 65109900167188257,2461735422517374779,99148196540813749081
%N A136727 E.g.f.: A(x) = (exp(x)/(3 - 2*exp(x)))^(1/3).
%C A136727 G.f. of variant A014307 is B(x) = sqrt(exp(x)/(2-exp(x))), which satisfies: B(x) = 1 + integral(B(x)^3*exp(-x)).
%H A136727 Harvey P. Dale, <a href="/A136727/b136727.txt">Table of n, a(n) for n = 0..350</a>
%F A136727 E.g.f. A(x) satisfies: A(x) = 1 + integral( A(x)^4 * exp(-x) ).
%F A136727 O.g.f.: 1/(1 - x/(1-2*x/(1 - 4*x/(1-4*x/(1 - 7*x/(1-6*x/(1 - 10*x/(1-8*x/(1 - 13*x/(1-10*x/(1 - ...))))))))))), a continued fraction.
%F A136727 G.f.: 1/G(0) where G(k) = 1 - x*(3*k+1)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction ). - _Sergei N. Gladkovskii_, Mar 23 2013
%F A136727 a(n) ~ n! * sqrt(3)*2^(2/3)*Gamma(2/3)/(4*Pi*n^(2/3)*(log(3/2))^(n+1/3)). - _Vaclav Kotesovec_, Jun 25 2013
%F A136727 a(n) = 1 + 2 * Sum_{k=1..n-1} (binomial(n,k) - 1) * a(k). - _Ilya Gutkovskiy_, Jul 09 2020
%F A136727 From _Seiichi Manyama_, Nov 15 2023: (Start)
%F A136727 a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (3*j+1)) * Stirling2(n,k).
%F A136727 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (2*k/n - 3) * binomial(n,k) * a(n-k).
%F A136727 a(0) = 1; a(n) = a(n-1) + 2*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)
%e A136727 E.g.f.: A(x) = 1 + x + 3/2*x^2 + 17/6*x^3 + 139/24*x^4 + 1481/120*x^5 +...
%t A136727 With[{nn=20},CoefficientList[Series[(Exp[x]/(3-2Exp[x]))^(1/3),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jan 26 2013 *)
%o A136727 (PARI) {a(n) = n!*polcoeff((exp(x +x*O(x^n))/(3-2*exp(x +x*O(x^n))))^(1/3),n)}
%o A136727 for(n=0,25,print1(a(n),", "))
%o A136727 (PARI) /* As solution to integral equation: */
%o A136727 {a(n) = local(A=1+x+x*O(x^n)); for(i=0,n, A = 1 + intformal(A^4*exp(-x+x*O(x^n)))); n!*polcoeff(A,n)}
%o A136727 for(n=0,25,print1(a(n),", "))
%Y A136727 Cf. A201339, variants: A014307, A136728, A136729.
%K A136727 nonn
%O A136727 0,3
%A A136727 _Paul D. Hanna_, Jan 24 2008