A187535 Central Lah numbers: a(n) = A105278(2*n,n) = A008297(2*n,n).

1, 2, 36, 1200, 58800, 3810240, 307359360, 29682132480, 3339239904000, 428906814336000, 61934143990118400, 9931984545324441600, 1751339941492209868800, 336796142594655744000000, 70149825129001153536000000, 15732267448930658699673600000

Offset: 0

Author

Emanuele Munarini, Mar 11 2011

Keywords

• nonn
• easy

Comments

a(n) is the number of Lah partitions of a set of size 2n with n blocks.

Crossrefs

Cf. A008297, A111596, A066667, A187536, A187538, A187539, A187540, A187542 - A187548, A090181, A125558.

Programs

Maple

A187535:= n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(A187535(n),n=0..12);

Mathematica

a[n_]:=If[n==0,1,Binomial[2n-1,n-1](2n)!/n!]
Table[a[n],{n,0,12}]
(* Alternative: *)
a[n_] := Binomial[2*n, n] FactorialPower[2*n - 1, n];
Table[a[n], {n, 0, 15}] (* Peter Luschny, Jun 15 2022 *)

Maxima

a(n) := if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(a(n),n,0,12);

Sage

[catalan_number(n)*binomial(2*n-1,n)*factorial(n+1) for n in range(15)] # Peter Luschny, Oct 07 2014

Formula

a(n) = binomial(2n-1,n-1)*(2n)!/n! (for n>0).

D-finite with recurrence (n+1)*a(n+1) = 4*(2n+1)^2*a(n) - delta(n,0).

a(n) ~ 2^(4*n)*n^n*exp(-n)/sqrt(2*n*Pi).

a(n)*a(n+2) - a(n+1)^2 is >= 0 and is a multiple of 2^(n+3) for all nonnegative n.

a(n) == 0 (mod 10) for n>3.

E.g.f.: 1/2 + K(16x)/Pi, where K(z) is the complete elliptic integral of the first kind, which can also be written as a Legendre function of the second kind.

a(n) = Catalan(n)*C(2*n-1,n)*(n+1)!. - Peter Luschny, Oct 07 2014

a(n) = A125558(n)*(n+1)! = A090181(2*n,n)*(n+1)!. - Peter Luschny, Oct 07 2014

a(n) = (2/n)*(Gamma(2*n)^2/Gamma(n)^3) for n>0. - Peter Luschny, Oct 17 2014