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

1, 2, 9, 60, 525, 5670, 72765, 1081080, 18243225, 344594250, 7202019825, 164991726900, 4111043861925, 110681950128750, 3201870700153125, 99044533658070000, 3262279327362680625, 113987877673731311250, 4211218814057295665625, 164015890652757831187500

Offset

0,2

Comments

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

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 166-167.

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)/(1-2*x)^(3/2) = d/dx (x/(1-2*x)^(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). - Sergei N. Gladkovskii, Sep 25 2012

From Peter Bala, Nov 07 2016 and May 14 2020: (Start)

a(n) = (n + 1)*(2*n - 1)/n * a(n-1) with a(0) = 1.

a(n) = 2*a(n-1) + (2*n - 3)*(2*n + 1)*a(n-2) with a(0) = 1, a(1) = 2.

(End)

a(n) = A059366(n+1, n) = A059366(n+1, 1). - Petros Hadjicostas, May 13 2020

Maple

f:= x-> x/sqrt(1-2*x): a:= n-> subs(x=0, (D@@(n+1))(f)(x)):
seq(a(n), n=0..20); # Zerinvary Lajos, Apr 04 2009
# second Maple program:
a:= n-> (n+1)*doublefactorial(2*n-1):
seq(a(n), n=0..23);  # Alois P. Heinz, May 13 2020

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)

Cf. A059366.

Sequence in context: A111558 A322943 A168449 * A161391 A120014 A036774

Adjacent sequences: A001190 A001191 A001192 * A001194 A001195 A001196

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Better description from Wouter Meeussen, Mar 08 2001

More terms from James A. Sellers, May 01 2000

Status

approved