0,2

Contribution from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 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+4)/(1-4*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). W. Lang, Oct 06 2008. (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]

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

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

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

E.g.f.: (1+x)/(1-2x)^(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 (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 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. W. Lang, Oct 06 2008. (End)

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]

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

Cf. A002544, A001814, A001876-A001878.

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

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

Equals row sums of A163938.

(End)

Sequence in context: A101600 A135148 A137974 this_sequence A019577 A097814 A084064

Adjacent sequences: A001876 A001877 A001878 this_sequence A001880 A001881 A001882

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

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