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%I M4251 N1775 #91 Jun 28 2023 19:14:01
%S 1,6,45,420,4725,62370,945945,16216200,310134825,6547290750,
%T 151242416325,3794809718700,102776096548125,2988412653476250,
%U 92854250304440625,3070380543400170000,107655217802968460625,3989575718580595893750,155815096120119939628125
%N a(n) = (2n+2)!/(n!*2^(n+1)).
%C From _Wolfdieter Lang_, Oct 06 2008: (Start)
%C 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.
%C 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)
%C 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
%C First differences of A193651. - _Vladimir Reshetnikov_, Apr 25 2016
%C a(n-2) is the number of maximal elements in the absolute order of the Coxeter group of type D_n. - _Jose Bastidas_, Nov 01 2021
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77 (Problem 10, values of Bessel polynomials).
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A001879/b001879.txt">Table of n, a(n) for n = 0..200</a>
%H Selden Crary, Richard Diehl Martinez and Michael Saunders, <a href="https://arxiv.org/abs/1707.00705">The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters</a>, arXiv:1707.00705 [stat.ME], 2017, Table 1.
%H Alexander Kreinin, <a href="https://www.researchgate.net/publication/294260037_Integer_Sequences_and_Laplace_Continued_Fraction">Integer Sequences and Laplace Continued Fraction</a>, Preprint 2016.
%H J. Riordan, <a href="/A001820/a001820.pdf">Notes to N. J. A. Sloane, Jul. 1968</a>
%F E.g.f.: (1+x)/(1-2*x)^(5/2).
%F 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
%F From _Wolfdieter Lang_, Oct 06 2008: (Start)
%F a(n) = (n+1)*(2*n+1)!! with the double factorials (2*n+1)!!=A001147(n+1).
%F D-finite with recurrence a(n) = 6*a(n-1) + ((2*n-1)^2)*a(n-2), a(-1)=0, a(0)=1. (End)
%F With interpolated 0's, e.g.f.: B(A(x)) where B(x)= x exp(x) and A(x)=x^2/2.
%F 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
%F 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
%F From _Karol A. Penson_, Jul 12 2013: (Start)
%F Integral representation as n-th moment of a signed function w(x) of bounded variation on (0,infinity),
%F w(x) = -(1/4)*sqrt(2)*sqrt(x)*(1-x)*exp(-x/2)/sqrt(Pi):
%F a(n) = Integral_{x>=0} x^n*w(x), n>=0.
%F For x>1, w(x)>0. w(0)=w(1)=limit(w(x),x=infinity)=0. For x<1, w(x)<0.
%F Asymptotics: a(n)->(1/576)*2^(1/2+n)*(1152*n^2+1680*n+505)*exp(-n)*(n)^(n), for n->infinity. (End)
%F G.f.: 2F0(3/2,2;;2x). - _R. J. Mathar_, Aug 08 2015
%p 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
%t Table[(2n+2)!/(n!2^(n+1)),{n,0,20}] (* _Vincenzo Librandi_, Nov 22 2011 *)
%o (PARI) a(n)=if(n<0,0,(2*n+2)!/n!/2^(n+1))
%o (Magma) [Factorial(2*n+2)/(Factorial(n)*2^(n+1)): n in [0..20]]; // _Vincenzo Librandi_, Nov 22 2011
%Y Cf. A002544, A001814, A001876, A001877, A001878.
%Y Second column of triangle A001497. Equals (A001147(n+1)-A001147(n))/2.
%Y Equals row sums of A163938.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Entry revised Aug 31 2004 (thanks to _Ralf Stephan_ and _Michael Somos_)
%E E.g.f. in comment line corrected by _Wolfdieter Lang_, Nov 21 2011
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