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A001879 (2n+2)!/(n!2^(n+1)).
(Formerly M4251 N1775)
12
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]

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

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).

Three term 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)

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: A227169 A135148 A137974 * 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 November 26 23:13 EST 2014. Contains 250152 sequences.