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A001193
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(n+1)(2n)!/(2^n*n!) (or (n+1)(2n-1)!!).
(Formerly M1944 N0770)
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5
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1, 2, 9, 60, 525, 5670, 72765, 1081080, 18243225, 344594250, 7202019825, 164991726900, 4111043861925, 110681950128750, 3201870700153125, 99044533658070000, 3262279327362680625, 113987877673731311250, 4211218814057295665625, 164015890652757831187500
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OFFSET
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0,2
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COMMENTS
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Solution to y'=A(x), y(0)=0 satisfies 0=x^2+2*y^2*x-y^2. - Michael Somos Mar 11 2004
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..100
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FORMULA
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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
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MAPLE
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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); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2009]
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MATHEMATICA
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Table[(n+1) (2*n-1)!!, {n, 0, 20}] (* From Vladimir Joseph Stephan Orlovsky, Apr 14 2011 *)
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PROG
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(PARI) a(n)=if(n<0, 0, (n+1)*(2*n)!/(2^n*n!))
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CROSSREFS
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Contribution 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: A120970 A111558 A168449 * A161391 A120014 A036774
Adjacent sequences: A001190 A001191 A001192 * A001194 A001195 A001196
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Better description from Wouter Meeussen, Mar 08 2001
More terms from James A. Sellers, May 01 2000
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STATUS
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approved
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