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A001879
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a(n) = (2n+2)!/(n!*2^(n+1)).
(Formerly M4251 N1775)
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21
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1, 6, 45, 420, 4725, 62370, 945945, 16216200, 310134825, 6547290750, 151242416325, 3794809718700, 102776096548125, 2988412653476250, 92854250304440625, 3070380543400170000, 107655217802968460625, 3989575718580595893750, 155815096120119939628125
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OFFSET
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0,2
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COMMENTS
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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
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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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
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) = Integral_{x>=0} x^n*w(x), 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)
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MAPLE
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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
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MATHEMATICA
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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