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 A001193 a(n) = (n+1)(2n)!/(2^n*n!) ( = (n+1)(2n-1)!!). (Formerly M1944 N0770) 7
 1, 2, 9, 60, 525, 5670, 72765, 1081080, 18243225, 344594250, 7202019825, 164991726900, 4111043861925, 110681950128750, 3201870700153125, 99044533658070000, 3262279327362680625, 113987877673731311250, 4211218814057295665625, 164015890652757831187500 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Solution to y'=A(x), y(0)=0 satisfies 0=x^2+2*y^2*x-y^2. - Michael Somos, Mar 11 2004 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 167. 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). LINKS T. D. Noe, Table of n, a(n) for n = 0..100 FORMULA E.g.f.: (1-x)/(1-2x)^(3/2) = d/dx x/(1-2x)^(1/2). a(n) = uppermost term in the vector (M(T))^n * [1,0,0,0,...] where T = Transpose, and M = the production matrix: 1, 2 1, 2, 3 1, 2, 3, 4 1, 2, 3, 4, 5 ... - Gary W. Adamson, Jul 08 2011 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 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 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 MAPLE 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 MATHEMATICA Table[(n+1) (2*n-1)!!, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Apr 14 2011 *) PROG (PARI) a(n)=if(n<0, 0, (n+1)*(2*n)!/(2^n*n!)) CROSSREFS From Johannes W. Meijer, Nov 12 2009: (Start) Equals the first right hand column of A167591. Equals the first left hand column of A167594. (End) Sequence in context: A111558 A322943 A168449 * A161391 A120014 A036774 Adjacent sequences:  A001190 A001191 A001192 * A001194 A001195 A001196 KEYWORD nonn,easy AUTHOR EXTENSIONS Better description from Wouter Meeussen, Mar 08 2001 More terms from James A. Sellers, May 01 2000 STATUS approved

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Last modified January 18 13:47 EST 2020. Contains 331009 sequences. (Running on oeis4.)