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A052296 Triangle read by rows: T(n,k) = number of labeled digraphs with n nodes and k arcs and without directed paths of length >=2, with 0 <= k <= floor(n^2/4).

%I A052296 #13 Feb 05 2024 09:44:41
%S A052296 1,1,1,2,1,6,6,1,12,36,32,6,1,20,120,280,280,120,20,1,30,300,1320,
%T A052296 2910,3492,2400,960,210,20,1,42,630,4480,17220,39144,56294,53760,
%U A052296 35070,15680,4662,840,70,1,56,1176,12320,73220,269136,654304,1108928,1362900
%N A052296 Triangle read by rows: T(n,k) = number of labeled digraphs with n nodes and k arcs and without directed paths of length >=2, with 0 <= k <= floor(n^2/4).
%F A052296 G.f. for n-th row: Sum_{k=0..n} binomial(n, k)*((1+x)^k-1)^(n-k). - _Vladeta Jovovic_, Apr 04 2003
%F A052296 E.g.f.: Sum_{n>=0} exp(y*((1+x)^n-1))*y^n/n!. - _Vladeta Jovovic_, May 28 2004
%F A052296 T(n,3) = n*(n-1)*(n-2)*(n-3)*(n^2-3*n+4)/6, n>=4. - _R. J. Mathar_, Mar 16 2021
%e A052296   1;
%e A052296   1;
%e A052296   1,  2;
%e A052296   1,  6,   6;
%e A052296   1, 12,  36,   32,    6;
%e A052296   1, 20, 120,  280,  280,  120,   20;
%e A052296   1, 30, 300, 1320, 2910, 3492, 2400, 960, 210, 20;
%e A052296   ...
%p A052296 A052296 := proc(n,k)
%p A052296     local x,l ;
%p A052296     add(binomial(n,l)*((1+x)^l-1)^(n-l),l=0..n) ;
%p A052296     expand(%) ;
%p A052296     coeftayl(%,x=0,k) ;
%p A052296 end proc: # _R. J. Mathar_, Mar 16 2021
%Y A052296 Row sums give A001831.
%Y A052296 Cf. A002378 (k=1), A083374 (k=2).
%K A052296 nonn,tabf
%O A052296 0,4
%A A052296 _Vladeta Jovovic_, Feb 08 2000