A102692 - OEIS

%I #17 Nov 18 2020 06:50:29

%S 1,2,4,28,336,5808,132000,3731040,126362880,4993309440,225677975040,

%T 11487263961600,650467886745600,40565803419187200,2763133948128153600,

%U 204127536266119065600,16257504491853520896000,1388699783389209747456000,126649067639795642253312000

%N a(n) = number of digraphs (allowing loops) with vertices 1,2,...,n that have a unique Eulerian tour (up to cyclic shift).

%D R. P. Stanley, unpublished work.

%H Alois P. Heinz, <a href="/A102692/b102692.txt">Table of n, a(n) for n = 0..349</a>

%F a(n) = (n-1)! * (b(n-1) + b(n)), where b(n) is a little Schroeder number (A001003).

%F a(n) ~ 2^(1/4) * (1 + sqrt(2))^(2*n) * n^(n-2) / exp(n). - _Vaclav Kotesovec_, May 22 2018

%e a(3) = 2!*(3+11) = 28. There are 16 such digraphs which are triangles with a possible loop at each vertex and 12 which consist of two 2-cycles with a common vertex and a possible loop at the other two vertices.

%p a:= proc(n) option remember; `if`(n<2, n+1,

%p       (n-1)*(6*(2*n^2-4*n+1)*a(n-1)-(n-2)*

%p       (n-3)*(2*n-1)*a(n-2))/((n+1)*(2*n-3)))

%p     end:

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, May 21 2018

%t b[n_] := Hypergeometric2F1[-n, n+1, 2, -1]/2;

%t a[n_] := Switch[n, 0, 1, 1, 2, _, (n-1)! (b[n-1] + b[n])];

%t a /@ Range[0, 20] (* _Jean-François Alcover_, Nov 18 2020 *)

%Y Cf. A001003.

%K nonn

%O 0,2

%A _Richard Stanley_, Feb 04 2005

%E a(0)=1 prepended and one typo corrected by _Alois P. Heinz_, May 21 2018