cp's OEIS Frontend

A057274 Triangle T(n,k) of the number of digraphs with a source on n labeled nodes with k arcs, k=0..n*(n-1).

%I A057274 #15 Jan 11 2022 19:53:47
%S A057274 1,0,2,1,0,0,9,20,15,6,1,0,0,0,64,330,720,914,792,495,220,66,12,1,0,0,
%T A057274 0,0,625,5804,24560,63940,117310,164260,183716,167780,125955,77520,
%U A057274 38760,15504,4845,1140,190,20,1
%N A057274 Triangle T(n,k) of the number of digraphs with a source on n labeled nodes with k arcs, k=0..n*(n-1).
%H A057274 Andrew Howroyd, <a href="/A057274/b057274.txt">Table of n, a(n) for n = 1..2680</a> (rows 1..20)
%H A057274 V. Jovovic and G. Kilibarda, <a href="http://dx.doi.org/10.1016/S0012-365X(00)00112-6">Enumeration of labeled quasi-initially connected digraphs</a>, Discrete Math., 224 (2000), 151-163.
%e A057274 Triangle begins:
%e A057274   1;
%e A057274   0, 2, 1;
%e A057274   0, 0, 9, 20,  15,   6,   1;
%e A057274   0, 0, 0, 64, 330, 720, 914, 792, 495, 220, 66, 12, 1;
%e A057274   ...
%e A057274 The number of digraphs with a source on 3 labeled nodes is the sum of the terms in row 3, i.e., 0+0+9+20+15+6+1 = 51 = A003028(3).
%o A057274 (PARI) \\ See A057273 for Strong.
%o A057274 Lambda(t, nn, e=2)={my(v=vector(1+nn)); for(n=0, nn, v[1+n] = e^(n*(n+t-1)) - sum(k=0, n-1, binomial(n,k)*e^((n-1)*(n-k))*v[1+k])); v}
%o A057274 Initially(n, e=2)={my(s=Strong(n, e), v=vector(n)); for(k=1, n, my(u=Lambda(k, n-k, e)); for(i=k, n, v[i] += binomial(i,k)*u[1+i-k]*s[k])); v }
%o A057274 row(n)={ Vecrev(Initially(n, 1+'y)[n]) }
%o A057274 { for(n=1, 5, print(row(n))) } \\ _Andrew Howroyd_, Jan 11 2022
%Y A057274 Row sums give A003028.
%Y A057274 The unlabeled version is A057277.
%Y A057274 Cf. A057271, A057272, A057273, A057275, A062735.
%K A057274 nonn,tabf
%O A057274 1,3
%A A057274 _Vladeta Jovovic_, Goran Kilibarda, Sep 14 2000