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A001193 a(n) = (n+1)*(2*n)!/(2^n*n!) = (n+1)*(2n-1)!!.

%I A001193 M1944 N0770 #64 Jul 02 2025 16:01:54
%S A001193 1,2,9,60,525,5670,72765,1081080,18243225,344594250,7202019825,
%T A001193 164991726900,4111043861925,110681950128750,3201870700153125,
%U A001193 99044533658070000,3262279327362680625,113987877673731311250,4211218814057295665625,164015890652757831187500
%N A001193 a(n) = (n+1)*(2*n)!/(2^n*n!) = (n+1)*(2n-1)!!.
%C A001193 Solution to y' = A(x), y(0) = 0 satisfies 0 = x^2 + 2*y^2*x - y^2, where A(x) = e.g.f. - _Michael Somos_, Mar 11 2004
%D A001193 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 166-167.
%D A001193 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001193 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001193 T. D. Noe, <a href="/A001193/b001193.txt">Table of n, a(n) for n = 0..100</a>
%F A001193 E.g.f.: (1-x)/(1-2*x)^(3/2) = d/dx (x/(1-2*x)^(1/2)).
%F A001193 a(n) = uppermost term in the vector (M(T))^n * [1,0,0,0,...], where T = Transpose and M = the production matrix:
%F A001193   1, 2;
%F A001193   1, 2, 3;
%F A001193   1, 2, 3, 4;
%F A001193   1, 2, 3, 4, 5;
%F A001193   ...
%F A001193 - _Gary W. Adamson_, Jul 08 2011
%F A001193 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 A001193 G.f.: U(0)/2  where U(k) = 1 + (2*k+1)/(1 - x/(x + 1/U(k+1))) (continued fraction). - _Sergei N. Gladkovskii_, Sep 25 2012
%F A001193 From _Peter Bala_, Nov 07 2016 and May 14 2020: (Start)
%F A001193 a(n) = (n + 1)*(2*n - 1)/n * a(n-1) with a(0) = 1.
%F A001193 a(n) = 2*a(n-1) + (2*n - 3)*(2*n + 1)*a(n-2) with a(0) = 1, a(1) = 2.
%F A001193 (End)
%F A001193 a(n) = A059366(n+1, n) = A059366(n+1, 1). - _Petros Hadjicostas_, May 13 2020
%p A001193 f:= x-> x/sqrt(1-2*x): a:= n-> subs(x=0, (D@@(n+1))(f)(x)):
%p A001193 seq(a(n), n=0..20); # _Zerinvary Lajos_, Apr 04 2009
%p A001193 # second Maple program:
%p A001193 a:= n-> (n+1)*doublefactorial(2*n-1):
%p A001193 seq(a(n), n=0..23);  # _Alois P. Heinz_, May 13 2020
%t A001193 Table[(n+1) (2*n-1)!!, {n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 14 2011 *)
%o A001193 (PARI) a(n)=if(n<0, 0, (n+1)*(2*n)!/(2^n*n!))
%Y A001193 From _Johannes W. Meijer_, Nov 12 2009: (Start)
%Y A001193 Equals the first right hand column of A167591.
%Y A001193 Equals the first left hand column of A167594. (End)
%Y A001193 Cf. A059366.
%K A001193 nonn,easy
%O A001193 0,2
%A A001193 _N. J. A. Sloane_
%E A001193 Better description from _Wouter Meeussen_, Mar 08 2001
%E A001193 More terms from _James Sellers_, May 01 2000