A350791 - OEIS

%I #12 Jan 22 2022 23:42:58

%S 1,0,2,0,0,6,6,0,0,0,24,132,180,84,12,0,0,0,0,120,1800,8000,16160,

%T 18180,12580,5560,1560,260,20,0,0,0,0,0,720,22320,214800,999450,

%U 2764650,5125380,6844380,6882150,5355750,3277200,1586520,605370,179250,39900,6300,630,30

%N Triangle read by rows: T(n,k) is the number of digraphs on n labeled nodes with k arcs and a global source and sink, n >= 1, k = 0..max(1,n-1)*(n-2)+1.

%H Andrew Howroyd, <a href="/A350791/b350791.txt">Table of n, a(n) for n = 1..2319</a> (rows 1..20)

%H R. W. Robinson, <a href="http://cobweb.cs.uga.edu/~rwr/publications/components.pdf">Counting digraphs with restrictions on the strong components</a>, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.

%e Triangle begins:

%e   [1] 1;

%e   [2] 0, 2;

%e   [3] 0, 0, 6, 6;

%e   [4] 0, 0, 0, 24, 132, 180, 84, 12;

%e   ...

%o (PARI) \\ Following Eqn 21 in the Robinson reference.

%o Z(p,f)={my(n=serprec(p,x)); serconvol(p, sum(k=0, n-1, x^k*f(k), O(x^n)))}

%o G(e,p)={Z(p, k->1/e^(k*(k-1)/2))}

%o U(e,p)={Z(p, k->e^(k*(k-1)/2))}

%o DigraphEgf(n,e)={sum(k=0, n, e^(k*(k-1))*x^k/k!, O(x*x^n) )}

%o StrongD(n,e=2)={-log(U(e, 1/G(e, DigraphEgf(n, e))))}

%o InitFinallyV(n, e=2)={my(S=StrongD(n, e)); Vec(serlaplace( x - x^2 + exp(S) * U(e, G(e, x*exp(-S))^2*G(e, DigraphEgf(n,e))) ))}

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

%o { for(n=1, 5, print(row(n))) }

%Y Row sums are A350790.

%Y The unlabeled version is A350795.

%Y Cf. A057271, A350793.

%K nonn,tabf

%O 1,3

%A _Andrew Howroyd_, Jan 16 2022