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%I #25 Apr 12 2023 09:00:16
%S 1,2,12,192,24,8160,2400,898560,384480,14400,245145600,126040320,
%T 9777600,50400,159035627520,90043269120,9660672000,179222400,80640,
%U 237882053283840,141969202744320,17961178152960,547498828800,2586608640,802369403419852800
%N Triangle read by rows: number of extensional acyclic digraphs on n labeled nodes with k sources.
%H S. Wagner, <a href="http://math.sun.ac.za/swagner/DigraphsFull.pdf">Asymptotic enumeration of extensional acyclic digraphs</a>, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).
%e Triangle begins:
%e 1;
%e 2;
%e 12;
%e 192, 24;
%e 8160, 2400;
%e 898560, 384480, 14400;
%e 245145600, 126040320, 9777600, 50400;
%e ...
%p 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
%t 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}]];
%t 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]];
%t Table[A182162[n, l], {n, 1, 10}, {l, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 09 2023, after _Nathaniel Johnston_ *)
%Y Row sums give A182161. First column is A182163. Row lengths are A182220.
%K nonn,tabf
%O 1,2
%A _N. J. A. Sloane_, Apr 15 2012
%E a(15)-a(25) from _Nathaniel Johnston_, Apr 18 2012
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