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

Author

Benoit Cloitre, Apr 04 2002

Keywords

• nonn
• easy

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
• Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
• Wojciech Mlotkowski and Anna Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.

Crossrefs

Cf. A043302, A144505.

Programs

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

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 *)

PARI

x='x+O('x^66); Vec(serlaplace(exp(2-2*sqrt(1-2*x))/sqrt(1-2*x))) \\ Joerg Arndt, May 04 2013

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

Extensions

Edited by Michael Somos, Jul 16 2002