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E.g.f.: A(x) = [ exp(x)/(3 - 2*exp(x)) ]^(1/3).
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%I #26 Nov 15 2023 08:03:41

%S 1,1,3,17,139,1481,19443,303297,5480219,112549881,2589274883,

%T 65957355377,1842897053099,56038776055081,1842278768795923,

%U 65109900167188257,2461735422517374779,99148196540813749081

%N E.g.f.: A(x) = [ exp(x)/(3 - 2*exp(x)) ]^(1/3).

%C 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 Harvey P. Dale, <a href="/A136727/b136727.txt">Table of n, a(n) for n = 0..350</a>

%F E.g.f. A(x) satisfies: A(x) = 1 + integral( A(x)^4 * exp(-x) ).

%F 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 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 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 a(n) = 1 + 2 * Sum_{k=1..n-1} (binomial(n,k) - 1) * a(k). - _Ilya Gutkovskiy_, Jul 09 2020

%F From _Seiichi Manyama_, Nov 15 2023: (Start)

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (3*j+1)) * Stirling2(n,k).

%F a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (2*k/n - 3) * binomial(n,k) * a(n-k).

%F a(0) = 1; a(n) = a(n-1) + 2*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)

%e E.g.f.: A(x) = 1 + x + 3/2*x^2 + 17/6*x^3 + 139/24*x^4 + 1481/120*x^5 +...

%t 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 (PARI) {a(n) = n!*polcoeff((exp(x +x*O(x^n))/(3-2*exp(x +x*O(x^n))))^(1/3),n)}

%o for(n=0,25,print1(a(n),", "))

%o (PARI) /* As solution to integral equation: */

%o {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 for(n=0,25,print1(a(n),", "))

%Y Cf. A201339, variants: A014307, A136728, A136729.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 24 2008