cp's OEIS Frontend

A138764 E.g.f. A(x) equals the inverse function of log(x)/(x + x^2).

%I A138764 #14 Oct 03 2020 08:28:35
%S A138764 1,2,16,248,5824,184512,7379200,356956672,20274442240,1322971320320,
%T A138764 97542692798464,8020249539919872,727662513046159360,
%U A138764 72215332738579824640,7782298855258810482688,905031449967822916026368
%N A138764 E.g.f. A(x) equals the inverse function of log(x)/(x + x^2).
%C A138764 Let r = radius of convergence of A(x), then:
%C A138764 r = 0.116689393840305520533609707610483991781804638898970699779...
%C A138764 A(r) = 1.835037067429188745641951736620284283425600418229813004773...
%C A138764 where A(r) and r satisfy:
%C A138764 A(r) = exp( (1 + A(r))/(1 + 2*A(r)) ) and r = 1/(A(r)*(1 + 2*A(r))).
%C A138764 Also, r = (1-t)^2/(t*(1+t)) where (t/(1-t))^(1+t) = e. [_Paul D. Hanna_, Oct 03 2020, derived from the asymptotic formula given by _Vaclav Kotesovec_].
%H A138764 Seiichi Manyama, <a href="/A138764/b138764.txt">Table of n, a(n) for n = 0..331</a>
%F A138764 E.g.f. satisfies: A(x) = exp( x*[A(x) + A(x)^2] ).
%F A138764 a(n) = Sum_{k=0..n} binomial(n,k)*(n+k+1)^(n-1) - _Vladeta Jovovic_, Mar 31 2008.
%F A138764 a(n) = A138860(n)*2^n.
%F A138764 E.g.f. satisfies: A( x/( exp(x) + exp(2*x) ) ) = exp(x).
%F A138764 a(n) ~ n^(n-1)*(1+r)^n*r^(n+1)/(sqrt(1+3*r)*(1-r)^(2*n+1)*exp(n)), where r = 0.6472709258412625... is the root of the equation (r/(1-r))^(1+r) = e. - _Vaclav Kotesovec_, Jun 15 2013
%e A138764 E.g.f. A(x) = 1 + 2x + 16x^2/2! + 248x^3/3! + 5824x^4/4! +...
%t A138764 Table[Sum[Binomial[n,k]*(n+k+1)^(n-1),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 15 2013 *)
%o A138764 (PARI) {a(n) = local(A=1+x); for(i=0,n, A=exp(x*(A+A^2 +x*O(x^n) ))); n! * polcoeff(A,n) }
%o A138764 (PARI) {a(n) = sum(k=0, n, binomial(n, k)*(n+k+1)^(n-1))}
%Y A138764 Cf. A138860.
%K A138764 nonn
%O A138764 0,2
%A A138764 _Paul D. Hanna_, Mar 29 2008