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A350791
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.
5
1, 0, 2, 0, 0, 6, 6, 0, 0, 0, 24, 132, 180, 84, 12, 0, 0, 0, 0, 120, 1800, 8000, 16160, 18180, 12580, 5560, 1560, 260, 20, 0, 0, 0, 0, 0, 720, 22320, 214800, 999450, 2764650, 5125380, 6844380, 6882150, 5355750, 3277200, 1586520, 605370, 179250, 39900, 6300, 630, 30
OFFSET
1,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..2319 (rows 1..20)
R. W. Robinson, Counting digraphs with restrictions on the strong components, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.
EXAMPLE
Triangle begins:
[1] 1;
[2] 0, 2;
[3] 0, 0, 6, 6;
[4] 0, 0, 0, 24, 132, 180, 84, 12;
...
PROG
(PARI) \\ Following Eqn 21 in the Robinson reference.
Z(p, f)={my(n=serprec(p, x)); serconvol(p, sum(k=0, n-1, x^k*f(k), O(x^n)))}
G(e, p)={Z(p, k->1/e^(k*(k-1)/2))}
U(e, p)={Z(p, k->e^(k*(k-1)/2))}
DigraphEgf(n, e)={sum(k=0, n, e^(k*(k-1))*x^k/k!, O(x*x^n) )}
StrongD(n, e=2)={-log(U(e, 1/G(e, DigraphEgf(n, e))))}
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))) ))}
row(n)={Vecrev(InitFinallyV(n, 1+'y)[n]) }
{ for(n=1, 5, print(row(n))) }
CROSSREFS
Row sums are A350790.
The unlabeled version is A350795.
Sequence in context: A257549 A112964 A368845 * A128613 A231063 A295216
KEYWORD
nonn,tabf
AUTHOR
Andrew Howroyd, Jan 16 2022
STATUS
approved