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A001879 a(n) = (2n+2)!/(n!2^(n+1)).
(Formerly M4251 N1775)
19
1, 6, 45, 420, 4725, 62370, 945945, 16216200, 310134825, 6547290750, 151242416325, 3794809718700, 102776096548125, 2988412653476250, 92854250304440625, 3070380543400170000, 107655217802968460625, 3989575718580595893750, 155815096120119939628125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Wolfdieter Lang, Oct 06 2008: (Start)

a(n) is the denominator of the n-th approximant to the continued fraction 1^2/(6+3^2/(6+5^2/(6+... for Pi-3. W. Lang, Oct 06 2008, after an e-mail from R. Rosenthal. Cf. A142970 for the corresponding numerators.

The e.g.f. g(x)=(1+x)/(1-2*x)^(5/2) satisfies (1-4*x^2)*g''(x) - 2*(8*x+3)*g'(x) -9*g(x) = 0 (from the three term recurrence given below). Also g(x)=hypergeom([2,3/2],[1],2*x). (End)

Number of descents in all fixed-point-free involutions of {1,2,...,2(n+1)}. A descent of a permutation p is a position i such that p(i) > p(i+1). Example: a(1)=6 because the fixed-point-free involutions 2143, 3412, and 4321 have 2, 1, and 3 descents, respectively. - Emeric Deutsch, Jun 05 2009

First differences of A193651. - Vladimir Reshetnikov, Apr 25 2016

REFERENCES

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77 (Problem 10, values of Bessel polynomials).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Selden Crary, Richard Diehl Martinez, Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 1.

Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.

J. Riordan, Notes to N. J. A. Sloane, Jul. 1968

FORMULA

E.g.f.: (1+x)/(1-2*x)^(5/2).

a(n)*n = a(n-1)*(2n+1)*(n+1); a(n) = a(n-1)*(2n+4)-a(n-2)*(2n-1), if n>0. - Michael Somos, Feb 25 2004

From Wolfdieter Lang, Oct 06 2008: (Start)

a(n) = (n+1)*(2*n+1)!! with the double factorials (2*n+1)!!=A001147(n+1).

D-finite with recurrence a(n) = 6*a(n-1) + ((2*n-1)^2)*a(n-2), a(-1)=0, a(0)=1. (End)

With interpolated 0's, e.g.f.: B(A(x)) where B(x)= x exp(x) and A(x)=x^2/2.

G.f.: - G(0)/2 where G(k) =  1 - (2*k+3)/(1 - x/(x - (k+1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 06 2012

G.f.: (1-x)/(2*x^2*Q(0)) - 1/(2*x^2), where Q(k)= 1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013

From Karol A. Penson, Jul 12 2013: (Start)

Integral representation as n-th moment of a signed function w(x) of bounded variation on (0,infinity),

  w(x) = -(1/4)*sqrt(2)*sqrt(x)*(1-x)*exp(-x/2)/sqrt(Pi):

  a(n) = int(x^n*w(x),x=0..infinity), n>=0.

  For x>1, w(x)>0. w(0)=w(1)=limit(w(x),x=infinity)=0. For x<1, w(x)<0.

Asymptotics: a(n)->(1/576)*2^(1/2+n)*(1152*n^2+1680*n+505)*exp(-n)*(n)^(n), for n->infinity. (End)

G.f.: 2F0(3/2,2;;2x). - R. J. Mathar, Aug 08 2015

MAPLE

restart: G(x):=(1-x)/(1-2*x)^(1/2): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1], x) od:x:=0:seq(f[n], n=2..20); # Zerinvary Lajos, Apr 04 2009

MATHEMATICA

Table[(2n+2)!/(n!2^(n+1)), {n, 0, 20}] (* Vincenzo Librandi, Nov 22 2011 *)

PROG

(PARI) a(n)=if(n<0, 0, (2*n+2)!/n!/2^(n+1))

(MAGMA) [Factorial(2*n+2)/(Factorial(n)*2^(n+1)): n in [0..20]]; // Vincenzo Librandi, Nov 22 2011

CROSSREFS

Cf. A002544, A001814, A001876-A001878.

Second column of triangle A001497. Equals (A001147(n+1)-A001147(n))/2.

Equals row sums of A163938.

Sequence in context: A135148 A137974 A291421 * A019577 A097814 A239910

Adjacent sequences:  A001876 A001877 A001878 * A001880 A001881 A001882

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Entry revised Aug 31 2004 (thanks to Ralf Stephan and Michael Somos)

E.g.f. in comment line corrected by Wolfdieter Lang, Nov 21 2011

STATUS

approved

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Last modified September 22 14:03 EDT 2020. Contains 337289 sequences. (Running on oeis4.)