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%I #55 Sep 08 2022 08:44:55
%S 1,3,13,77,591,5627,64261,857901,13125559,226566107,4357258269,
%T 92408688077,2142828858847,53940356223483,1464960933469429,
%U 42699628495507373,1329548327094606279,44045893308104036699,1546924459092019709581,57412388559637145401293
%N a(n) = 2^n*Sum_{k=0..n} (n+k)!/((n-k)!*k!*4^k).
%D Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; see Integrals and Asymptotic Expansions, p. 229.
%D I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed., Section 3.737.1, p. 423.
%H Vincenzo Librandi, <a href="/A043301/b043301.txt">Table of n, a(n) for n = 0..200</a>
%H W. Mlotkowski, A. Romanowicz, <a href="http://www.math.uni.wroc.pl/~pms/files/33.2/Article/33.2.19.pdf">A family of sequences of binomial type</a>, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
%F D-finite with recurrence: a(n) = (2*n-1)*a(n-1) + 4*a(n-2), n>1.
%F a(n) = 2^(n+1)n!(e^2/Pi)*Integral_{t=0..infinity} cos(2t)/(1+t^2)^(n+1)dt.
%F E.g.f.: 2*(e^2/Pi)*Integral_{t=0..infinity} cos(2t)/(1+t^2-2x)dt.
%F 2^n * y_n(1/2), where y_n(x) are the Bessel polynomials A001498.
%F G.f.: 1/G(0) where G(k) = 1 - 2*x - x*(k+1)/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Dec 17 2011
%F E.g.f.: exp(2-2*sqrt(1-2*x))/sqrt(1-2*x). - _Vaclav Kotesovec_, Oct 21 2012
%F a(n) ~ 2^(n+1/2)*n^n/exp(n-2). - _Vaclav Kotesovec_, Oct 21 2012
%F G.f.: T(0)/(1-2*x), where T(k) = 1 - x*(k+1)/( x*(k+1) - (1-2*x)^2/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 27 2013
%F a(n) = 2^(n+1)*exp(2)/sqrt(Pi)*BesselK(1/2+n,2). - _Gerry Martens_, Jul 22 2015
%F a(n) = 2^n*hypergeom( [n+1, -n], [], -1/4). - _Peter Luschny_, Nov 10 2016
%p f:= gfun:-rectoproc({a(0)=1, a(1)=3, a(n) = (2*n-1)*a(n-1) + 4*a(n-2)}, a(n), remember):
%p map(f, [$0..30]); # _Robert Israel_, Jul 23 2015
%p A043301 := n-> 2^n*hypergeom([n+1, -n], [], -1/4):
%p seq(simplify(A043301(n)), n=0..19); # _Peter Luschny_, Nov 10 2016
%t Table[2^n Sum[(n+k)!/((n-k)!k! 4^k),{k,0,n}],{n,0,20}] (* or *) RecurrenceTable[{a[0]==1,a[1]==3,a[n]==(2n-1)a[n-1]+4a[n-2]}, a[n], {n,20}] (* _Harvey P. Dale_, Aug 14 2011 *)
%t CoefficientList[Series[E^(2-2*Sqrt[1-2*x])/Sqrt[1-2*x],{x,0,20}],x]*Range[0,20]! (* _Vaclav Kotesovec_, Oct 21 2012 *)
%o (PARI) x='x+O('x^66); Vec(serlaplace(exp(2-2*sqrt(1-2*x))/sqrt(1-2*x))) \\ _Joerg Arndt_, May 04 2013
%o (Magma) I:=[3,13]; [1] cat [n le 2 select I[n] else (2*n-1)*Self(n-1) + 4*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jul 24 2015
%Y Cf. A043302, A144505.
%K nonn,easy
%O 0,2
%A _Benoit Cloitre_, Apr 04 2002
%E Edited by _Michael Somos_, Jul 16 2002
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