OFFSET
0,2
COMMENTS
a(n) is the number of Lah partitions of a set of size 2n with n blocks.
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) = (2/n)*(Gamma(2*n)^2/Gamma(n)^3) for n>0. - Peter Luschny, Oct 17 2014
MAPLE
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emanuele Munarini, Mar 11 2011
STATUS
approved