A043301 a(n) = 2^n*Sum_{k=0..n} (n+k)!/((n-k)!*k!*4^k).

1, 3, 13, 77, 591, 5627, 64261, 857901, 13125559, 226566107, 4357258269, 92408688077, 2142828858847, 53940356223483, 1464960933469429, 42699628495507373, 1329548327094606279, 44045893308104036699, 1546924459092019709581, 57412388559637145401293

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

Cf. A043302, A144505.

Sequence in context: A351421 A273953 A127127 * A141762 A062872 A288954

Adjacent sequences: A043298 A043299 A043300 * A043302 A043303 A043304

Keyword

nonn,easy

Author

Benoit Cloitre, Apr 04 2002

Extensions

Edited by Michael Somos, Jul 16 2002

Status

approved