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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). 7

%I

%S 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 0,0,625,5804,24560,63940,117310,164260,183716,167780,125955,77520,

%U 38760,15504,4845,1140,190,20,1

%N 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 Andrew Howroyd, <a href="/A057274/b057274.txt">Table of n, a(n) for n = 1..2680</a> (rows 1..20)

%H 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 Triangle begins:

%e 1;

%e 0, 2, 1;

%e 0, 0, 9, 20, 15, 6, 1;

%e 0, 0, 0, 64, 330, 720, 914, 792, 495, 220, 66, 12, 1;

%e ...

%e 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 (PARI) \\ See A057273 for Strong.

%o 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 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 row(n)={ Vecrev(Initially(n, 1+'y)[n]) }

%o { for(n=1, 5, print(row(n))) } \\ _Andrew Howroyd_, Jan 11 2022

%Y Row sums give A003028.

%Y The unlabeled version is A057277.

%Y Cf. A057271, A057272, A057273, A057275, A062735.

%K nonn,tabf,changed

%O 1,3

%A _Vladeta Jovovic_, Goran Kilibarda, Sep 14 2000

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Last modified January 24 19:34 EST 2022. Contains 350565 sequences. (Running on oeis4.)