OFFSET
0,2
REFERENCES
Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; see Integrals and Asymptotic Expansions, p. 229.
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed., Section 3.737.1, p. 423.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
W. Mlotkowski, A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
FORMULA
D-finite with recurrence: a(n) = (2*n-1)*a(n-1) + 4*a(n-2), n>1.
a(n) = 2^(n+1)n!(e^2/Pi)*Integral_{t=0..infinity} cos(2t)/(1+t^2)^(n+1)dt.
E.g.f.: 2*(e^2/Pi)*Integral_{t=0..infinity} cos(2t)/(1+t^2-2x)dt.
2^n * y_n(1/2), where y_n(x) are the Bessel polynomials A001498.
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
E.g.f.: exp(2-2*sqrt(1-2*x))/sqrt(1-2*x). - Vaclav Kotesovec, Oct 21 2012
a(n) ~ 2^(n+1/2)*n^n/exp(n-2). - Vaclav Kotesovec, Oct 21 2012
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
a(n) = 2^(n+1)*exp(2)/sqrt(Pi)*BesselK(1/2+n,2). - Gerry Martens, Jul 22 2015
a(n) = 2^n*hypergeom( [n+1, -n], [], -1/4). - Peter Luschny, Nov 10 2016
MAPLE
f:= gfun:-rectoproc({a(0)=1, a(1)=3, a(n) = (2*n-1)*a(n-1) + 4*a(n-2)}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Jul 23 2015
A043301 := n-> 2^n*hypergeom([n+1, -n], [], -1/4):
seq(simplify(A043301(n)), n=0..19); # Peter Luschny, Nov 10 2016
MATHEMATICA
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 *)
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 *)
PROG
(PARI) x='x+O('x^66); Vec(serlaplace(exp(2-2*sqrt(1-2*x))/sqrt(1-2*x))) \\ Joerg Arndt, May 04 2013
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Apr 04 2002
EXTENSIONS
Edited by Michael Somos, Jul 16 2002
STATUS
approved