A003409 a(n) = 3*binomial(2n-1,n). (Formerly M2814)

3, 9, 30, 105, 378, 1386, 5148, 19305, 72930, 277134, 1058148, 4056234, 15600900, 60174900, 232676280, 901620585, 3500409330, 13612702950, 53017895700, 206769793230, 807386811660, 3156148445580, 12350146091400, 48371405524650, 189615909656628, 743877799422156

Offset

1,1

References

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 = 1..200

C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.

C. Domb & A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358. (Annotated scanned copy)

C. Domb & A. J. Barrett, Notes on Table 2 in "Enumeration of ladder graphs", Discrete Math. 9 (1974), 55. (Annotated scanned copy)

Formula

a(n) = (3/2)*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015

From Stefano Spezia, Jul 05 2021: (Start)

O.g.f.: 6*x/((1 - sqrt(1 - 4*x))*sqrt(1 - 4*x)) - 3.

E.g.f.: 3*(exp(2*x)*I_0(2*x) - 1)/2, where I_n(x) is the modified Bessel function of the first kind.

a(n) ~ 3*4^n/(2*sqrt(n*Pi)). (End)

Maple

a := n -> (3/2)*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(a(n), n=1..26); # Peter Luschny, Dec 14 2015

Mathematica

Table[3*Binomial[2*n - 1, n], {n, 20}] (* T. D. Noe, Oct 07 2013 *)

Prog

(PARI) a(n) = 3*binomial(2*n-1, n) \\ Charles R Greathouse IV, Oct 23 2023

Crossrefs

Equals 3 * A001700.

Sequence in context: A339036 A360715 A029651 * A316371 A181933 A148957

Adjacent sequences: A003406 A003407 A003408 * A003410 A003411 A003412

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Jon E. Schoenfield, Mar 26 2010

Status

approved