A003409 - OEIS

%I M2814 #34 Oct 23 2023 12:38:47

%S 3,9,30,105,378,1386,5148,19305,72930,277134,1058148,4056234,15600900,

%T 60174900,232676280,901620585,3500409330,13612702950,53017895700,

%U 206769793230,807386811660,3156148445580,12350146091400,48371405524650,189615909656628,743877799422156

%N a(n) = 3*binomial(2n-1,n).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A003409/b003409.txt">Table of n, a(n) for n = 1..200</a>

%H C. Domb and A. J. Barrett, <a href="http://dx.doi.org/10.1016/0012-365X(74)90081-8">Enumeration of ladder graphs</a>, Discrete Math. 9 (1974), 341-358.

%H C. Domb & A. J. Barrett, <a href="/A003408/a003408.pdf">Enumeration of ladder graphs</a>, Discrete Math. 9 (1974), 341-358. (Annotated scanned copy)

%H C. Domb & A. J. Barrett, <a href="/A001764/a001764.pdf">Notes on Table 2 in "Enumeration of ladder graphs"</a>, Discrete Math. 9 (1974), 55. (Annotated scanned copy)

%F a(n) = (3/2)*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(1+n)). - _Peter Luschny_, Dec 14 2015

%F From _Stefano Spezia_, Jul 05 2021: (Start)

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

%F 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.

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

%p a := n -> (3/2)*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(1+n)):

%p seq(a(n), n=1..26); # _Peter Luschny_, Dec 14 2015

%t Table[3*Binomial[2*n - 1, n], {n, 20}] (* _T. D. Noe_, Oct 07 2013 *)

%o (PARI) a(n) = 3*binomial(2*n-1,n) \\ _Charles R Greathouse IV_, Oct 23 2023

%Y Equals 3 * A001700.

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Jon E. Schoenfield_, Mar 26 2010