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

%I A187535 #23 Jun 15 2022 11:46:34
%S A187535 1,2,36,1200,58800,3810240,307359360,29682132480,3339239904000,
%T A187535 428906814336000,61934143990118400,9931984545324441600,
%U A187535 1751339941492209868800,336796142594655744000000,70149825129001153536000000,15732267448930658699673600000
%N A187535 Central Lah numbers: a(n) = A105278(2*n,n) = A008297(2*n,n).
%C A187535 a(n) is the number of Lah partitions of a set of size 2n with n blocks.
%F A187535 a(n) = binomial(2n-1,n-1)*(2n)!/n! (for n>0).
%F A187535 D-finite with recurrence (n+1)*a(n+1) = 4*(2n+1)^2*a(n) - delta(n,0).
%F A187535 a(n) ~ 2^(4*n)*n^n*exp(-n)/sqrt(2*n*Pi).
%F A187535 a(n)*a(n+2) - a(n+1)^2 is >= 0 and is a multiple of 2^(n+3) for all nonnegative  n.
%F A187535 a(n) == 0 (mod 10) for n>3.
%F A187535 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.
%F A187535 a(n) = Catalan(n)*C(2*n-1,n)*(n+1)!. - _Peter Luschny_, Oct 07 2014
%F A187535 a(n) = A125558(n)*(n+1)! = A090181(2*n,n)*(n+1)!. - _Peter Luschny_, Oct 07 2014
%F A187535 a(n) = (2/n)*(Gamma(2*n)^2/Gamma(n)^3) for n>0. - _Peter Luschny_, Oct 17 2014
%p A187535 A187535:= n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
%p A187535 seq(A187535(n),n=0..12);
%t A187535 a[n_]:=If[n==0,1,Binomial[2n-1,n-1](2n)!/n!]
%t A187535 Table[a[n],{n,0,12}]
%t A187535 (* Alternative: *)
%t A187535 a[n_] := Binomial[2*n, n] FactorialPower[2*n - 1, n];
%t A187535 Table[a[n], {n, 0, 15}] (* _Peter Luschny_, Jun 15 2022 *)
%o A187535 (Maxima) a(n) := if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
%o A187535 makelist(a(n),n,0,12);
%o A187535 (Sage)
%o A187535 [catalan_number(n)*binomial(2*n-1,n)*factorial(n+1) for n in range(15)] # _Peter Luschny_, Oct 07 2014
%Y A187535 Cf. A008297, A111596, A066667, A187536, A187538, A187539, A187540, A187542 - A187548, A090181, A125558.
%K A187535 nonn,easy
%O A187535 0,2
%A A187535 _Emanuele Munarini_, Mar 11 2011