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A124212 Expansion of e.g.f. exp(x)/sqrt(2-exp(2*x)).

%I A124212 #31 Nov 18 2023 08:36:07
%S A124212 1,2,8,56,560,7232,114368,2139776,46223360,1132124672,30999600128,
%T A124212 938366468096,31114518056960,1121542540992512,43664751042265088,
%U A124212 1826043989622358016,81635676596544143360
%N A124212 Expansion of e.g.f. exp(x)/sqrt(2-exp(2*x)).
%H A124212 Robert Israel, <a href="/A124212/b124212.txt">Table of n, a(n) for n = 0..60</a>
%F A124212 E.g.f. satisfies: A'(x) = A(x) + A(x)^3 with A(0)=1. [From _Paul D. Hanna_, Oct 04 2008]
%F A124212 G.f.: 1/G(0) where G(k) = 1 - x*(4*k+2)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction ). - _Sergei N. Gladkovskii_, Mar 23 2013
%F A124212 G.f.: 2/G(0), where G(k)= 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 4*x*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 30 2013
%F A124212 a(n) ~ 2^(n+1/2)*n^n/(log(2)^(n+1/2)*exp(n)). - _Vaclav Kotesovec_, Jun 03 2013
%F A124212 From _Peter Bala_, Aug 30 2016: (Start)
%F A124212 a(n) = 1/sqrt(2) * Sum_{k >= 0} (1/8)^k*binomial(2*k,k)*(2*k + 1)^n = 1/sqrt(2) * Sum_{k >= 0} (-1/2)^k*binomial(-1/2,k)*(2*k + 1)^n. Cf. A176785, A124214 and A229558.
%F A124212 a(n) = Sum_{k = 0..n} (1/4)^k*binomial(2*k,k)*A145901(n,k).
%F A124212 a(n) = Sum_{k = 0..n} ( Sum_{i = 0..k} (-1)^(k-i)/4^k* binomial(2*k,k)*binomial(k,i)*(2*i + 1)^n ). (End)
%F A124212 a(n) = 2^n * A014307(n). - _Seiichi Manyama_, Nov 18 2023
%p A124212   N:= 60; # to get a(n) for n <= N
%p A124212 S:= series(exp(x)/sqrt(2-exp(2*x)), x, N+1):
%p A124212 seq(coeff(S,x,j), j=0..N); # _Robert Israel_, May 19 2014
%t A124212 CoefficientList[Series[E^x/Sqrt[2-E^(2*x)]-1, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 03 2013 *)
%o A124212 (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0,n,A=1+intformal(A+A^3)); n!*polcoeff(A,n)} \\ _Paul D. Hanna_, Oct 04 2008
%Y A124212 Cf. A000629, A000984, A014307, A124214, A145901, A176785, A186695, A229558.
%Y A124212 Cf. A276371, A365777.
%K A124212 nonn,easy
%O A124212 0,2
%A A124212 _Karol A. Penson_, Oct 19 2006
%E A124212 Definition corrected by _Robert Israel_, May 19 2014