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A001193 a(n) = (n+1)(2n)!/(2^n*n!) ( = (n+1)(2n-1)!!).
(Formerly M1944 N0770)
7

%I M1944 N0770

%S 1,2,9,60,525,5670,72765,1081080,18243225,344594250,7202019825,

%T 164991726900,4111043861925,110681950128750,3201870700153125,

%U 99044533658070000,3262279327362680625,113987877673731311250,4211218814057295665625,164015890652757831187500

%N a(n) = (n+1)(2n)!/(2^n*n!) ( = (n+1)(2n-1)!!).

%C Solution to y'=A(x), y(0)=0 satisfies 0=x^2+2*y^2*x-y^2. - _Michael Somos_, Mar 11 2004

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 167.

%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 T. D. Noe, <a href="/A001193/b001193.txt">Table of n, a(n) for n = 0..100</a>

%F E.g.f.: (1-x)/(1-2x)^(3/2) = d/dx x/(1-2x)^(1/2).

%F a(n) = uppermost term in the vector (M(T))^n * [1,0,0,0,...] where T = Transpose, and M = the production matrix:

%F 1, 2

%F 1, 2, 3

%F 1, 2, 3, 4

%F 1, 2, 3, 4, 5

%F ...

%F - _Gary W. Adamson_, Jul 08 2011

%F G.f.: A(x)=1 + 2*x/(G(0) -2*x) ; G(k) =1 + k + x*(k+2)*(2*k+1) - x*(k+1)*(k+3)*(2*k+3)/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Dec 06 2011

%F G.f.: U(0)/2 where U(k)= 1 + (2*k+1)/(1 - x/(x + 1/U(k+1))) ; (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Sep 25 2012

%F a(n) = (n + 1)*(2*n - 1)/n * a(n-1). a(n) = 2*a(n) + (2*n - 3)*(2*n + 1)*a(n-1). - _Peter Bala_, Nov 07 2016

%p restart: G(x):=(1-x)/(1-2*x)^(3/2): 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=0..17); # _Zerinvary Lajos_, Apr 04 2009

%t Table[(n+1) (2*n-1)!!,{n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 14 2011 *)

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

%Y From _Johannes W. Meijer_, Nov 12 2009: (Start)

%Y Equals the first right hand column of A167591.

%Y Equals the first left hand column of A167594.

%Y (End)

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E Better description from _Wouter Meeussen_, Mar 08 2001

%E More terms from _James A. Sellers_, May 01 2000

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Last modified August 18 16:26 EDT 2018. Contains 313833 sequences. (Running on oeis4.)