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A189054 Expansion of e.g.f. exp(x/sqrt(1-4*x^2)).

%I A189054 #25 Sep 03 2025 04:19:54
%S A189054 1,1,1,13,49,841,6001,126421,1371553,34081489,503678881,14391006301,
%T A189054 271223253841,8751666000793,201326507146129,7238365225056421,
%U A189054 197024810845531201,7810072695945382561,245787442777437613633
%N A189054 Expansion of e.g.f. exp(x/sqrt(1-4*x^2)).
%H A189054 Vincenzo Librandi, <a href="/A189054/b189054.txt">Table of n, a(n) for n = 0..200</a>
%F A189054 a(n) = n! * Sum_{k=0..n} (binomial((n-2)/2, (n-k)/2) * 2^(n-k-1) * ((-1)^(n-k)+1))/k!.
%F A189054 a(n) ~ (2*n)^(n-1/3) / (sqrt(3)*exp(n-3/4*(2*n)^(1/3))). - _Vaclav Kotesovec_, Jun 02 2013
%t A189054 CoefficientList[Series[Exp[x/Sqrt[1-4*x^2]], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 02 2013 *)
%o A189054 (Maxima)
%o A189054 a(n):= n!*sum((binomial((n-2)/2,(n-k)/2)*2^(n-k-1)*((-1)^(n-k)+1))/k!, k,0,n);
%o A189054 (PARI) x='x+O('x^66); /* that many terms */
%o A189054 egf=exp(x/sqrt(1-4*x^2)) /* = 1 +x +1/2*x^2 +13/6*x^3 +49/24*x^4 +... */
%o A189054 Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 22 2011 */
%Y A189054 Cf. A012150.
%K A189054 nonn,changed
%O A189054 0,4
%A A189054 _Vladimir Kruchinin_, Apr 16 2011