A182162 Triangle read by rows: number of extensional acyclic digraphs on n labeled nodes with k sources.

1, 2, 12, 192, 24, 8160, 2400, 898560, 384480, 14400, 245145600, 126040320, 9777600, 50400, 159035627520, 90043269120, 9660672000, 179222400, 80640, 237882053283840, 141969202744320, 17961178152960, 547498828800, 2586608640, 802369403419852800

Offset: 1

Author

N. J. A. Sloane, Apr 15 2012

Keywords

• nonn
• tabf

Examples

			Triangle begins:
          1;
          2;
         12;
        192,        24;
       8160,      2400;
     898560,    384480,   14400;
  245145600, 126040320, 9777600, 50400;
  ...
		

Links

• S. Wagner, Asymptotic enumeration of extensional acyclic digraphs, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).

Crossrefs

Row sums give A182161. First column is A182163. Row lengths are A182220.

Programs

Maple

A001192 := proc(n) option remember: if(n=0)then return 1: fi: return add((-1)^(n-k-1)*binomial(2^k-k,n-k)*procname(k), k=0..n-1); end: A182162 := proc(n,l) local vl: vl := add((-1)^(k-l)*binomial(n,k)*binomial(k,l)*binomial(2^(n-k)-n+k,k)*k!*(n-k)!*A001192(n-k), k=l..n): if(vl = 0)then return NULL: fi: return vl: end: for n from 1 to 10 do seq(A182162(n,l), l=1..n); od; # Nathaniel Johnston, Apr 18 2012

Mathematica

A001192[n_] := A001192[n] = If[n == 0, 1, Sum[(-1)^(n - k - 1)*Binomial[2^k - k, n - k]*A001192[k], {k, 0, n - 1}]];
A182162[n_, l_] := Module[{vl}, vl = Sum[(-1)^(k - l)* Binomial[n, k]*Binomial[k, l]*Binomial[2^(n - k) - n + k, k]*k!*(n - k)!*A001192[n - k], {k, l, n}]; If[vl == 0, Nothing, vl]];
Table[A182162[n, l], {n, 1, 10}, {l, 1, n}] // Flatten (* Jean-François Alcover, Mar 09 2023, after Nathaniel Johnston *)

Extensions

a(15)-a(25) from Nathaniel Johnston, Apr 18 2012