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

1, 2, 13, 132, 1861, 33630, 741721, 19318376, 580293001, 19749280410, 751052948581, 31563973120812, 1452693816505933, 72666254798417462, 3925430452931048881, 227747632524799252560, 14124278646990484707601

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..365

Formula

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).

E.g.f.: cosh((1-sqrt(1-4*x))/2)/sqrt(1-4*x). - Vladimir Kruchinin, May 03 2016

a(n) = Sum_{k=0..n/2}((2*n-2*k)!/((n-2*k)!*(2*k)!)). - Vladimir Kruchinin, May 03 2016

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

a(n) = (hypergeom([-n,n+1],[],-1)+(-1)^n*hypergeom([-n,n+1],[],1))/2. - Peter Luschny, May 03 2016

Example

a(3) = (4*3-2)*a(2)+a(1) = 10*13+2 = 132.

Maple

f:= gfun:-rectoproc({a(n)=(4*n-2)*a(n-1)+a(n-2), a(0)=1, a(1)=2}, a(n), remember):
map(f, [$0..50]); # Robert Israel, May 03 2016

Mathematica

a[n_] := Sum[(2n-2k)!/((n-2k)! (2k)!), {k, 0, n/2}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 10 2019, after Vladimir Kruchinin *)

Prog

(Maxima)
a(n):=sum((2*n-2*k)!/((n-2*k)!*(2*k)!), k, 0, n/2); /* Vladimir Kruchinin, May 03 2016 */

Crossrefs

Cf. A001517, A002119, A079166.

Sequence in context: A107721 A085882 A041509 * A223075 A239748 A355113

Adjacent sequences: A079162 A079163 A079164 * A079166 A079167 A079168

Keyword

nonn

Author

Henry Bottomley, Dec 31 2002

Status

approved