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A001879
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(2n+2)!/(n!2^(n+1)).
(Formerly M4251 N1775)
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10
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1, 6, 45, 420, 4725, 62370, 945945, 16216200, 310134825, 6547290750, 151242416325, 3794809718700, 102776096548125, 2988412653476250, 92854250304440625, 3070380543400170000, 107655217802968460625, 3989575718580595893750, 155815096120119939628125
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Contribution 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. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 05 2009]
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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).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
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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
Contribution 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
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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); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2009]
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MATHEMATICA
| Table[(2n+2)!/(n!2^(n+1)), {n, 0, 20}] (* Vincenzo Librandi, Nov 22 2011 *)
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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
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CROSSREFS
| Cf. A002544, A001814, A001876-A001878.
Second column of triangle A001497. Equals (A001147(n+1)-A001147(n))/2.
Equals row sums of A163938. - Johannes W. Meijer, Oct 16 2009
Sequence in context: A101600 A135148 A137974 * A019577 A097814 A084064
Adjacent sequences: A001876 A001877 A001878 * A001880 A001881 A001882
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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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.
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