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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

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.

Programs

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 *)

PARI

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

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

Extensions

Better description from Wouter Meeussen, Mar 08 2001

More terms from James Sellers, May 01 2000