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A079165 a(n) = (4n-2)*a(n-1)+a(n-2) with a(0)=1 and a(1)=2.

%I A079165 #19 Apr 10 2019 08:35:06
%S A079165 1,2,13,132,1861,33630,741721,19318376,580293001,19749280410,
%T A079165 751052948581,31563973120812,1452693816505933,72666254798417462,
%U A079165 3925430452931048881,227747632524799252560,14124278646990484707601
%N A079165 a(n) = (4n-2)*a(n-1)+a(n-2) with a(0)=1 and a(1)=2.
%H A079165 Robert Israel, <a href="/A079165/b079165.txt">Table of n, a(n) for n = 0..365</a>
%F A079165 a(n) = (A001517(n)+|A002119(n)|)/2 = A079166(2, n). a(n)/|A002119(n)| tends to 1.8591409...=(e+1)/2; a(n)/A001517(n) tends to 0.68393972...=2e/(e+1).
%F A079165 E.g.f.: cosh((1-sqrt(1-4*x))/2)/sqrt(1-4*x). - _Vladimir Kruchinin_, May 03 2016
%F A079165 a(n) = Sum_{k=0..n/2}((2*n-2*k)!/((n-2*k)!*(2*k)!)). - _Vladimir Kruchinin_, May 03 2016
%F A079165 a(n) = ((-1)^n*sqrt(Pi*exp(-1))*BesselI((2*n+1)/2, 1/2))/2 + (BesselK((2*n+1)/2, 1/2)*cosh(1/2))/sqrt(Pi), where BesselI(n,x) is the modified Bessel function of the first kind, BesselK(n,x) is the modified Bessel function of the second kind. - _Ilya Gutkovskiy_, May 03 2016
%F A079165 a(n) = (hypergeom([-n,n+1],[],-1)+(-1)^n*hypergeom([-n,n+1],[],1))/2. - _Peter Luschny_, May 03 2016
%e A079165 a(3) = (4*3-2)*a(2)+a(1) = 10*13+2 = 132.
%p A079165 f:= gfun:-rectoproc({a(n)=(4*n-2)*a(n-1)+a(n-2),a(0)=1,a(1)=2},a(n),remember):
%p A079165 map(f, [$0..50]); # _Robert Israel_, May 03 2016
%t A079165 a[n_] := Sum[(2n-2k)!/((n-2k)! (2k)!), {k, 0, n/2}];
%t A079165 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 10 2019, after _Vladimir Kruchinin_ *)
%o A079165 (Maxima)
%o A079165 a(n):=sum((2*n-2*k)!/((n-2*k)!*(2*k)!),k,0,n/2); /* _Vladimir Kruchinin_, May 03 2016 */
%Y A079165 Cf. A001517, A002119, A079166.
%K A079165 nonn
%O A079165 0,2
%A A079165 _Henry Bottomley_, Dec 31 2002