A361455 Triangle read by rows: T(n,k) is the number of simple digraphs on labeled n nodes with k strongly connected components.

1, 0, 1, 0, 1, 3, 0, 18, 21, 25, 0, 1606, 1173, 774, 543, 0, 565080, 271790, 122595, 59830, 29281, 0, 734774776, 229224750, 70500705, 25349355, 10110735, 3781503, 0, 3523091615568, 685793359804, 138122171880, 35130437825, 11002159455, 3767987307, 1138779265

Offset: 0

Author

Andrew Howroyd, Mar 16 2023

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  0,         1;
  0,         1,         3;
  0,        18,        21,       25;
  0,      1606,      1173,      774,      543;
  0,    565080,    271790,   122595,    59830,    29281;
  0, 734774776, 229224750, 70500705, 25349355, 10110735, 3781503;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50).

Crossrefs

Column k=1 is A003030.

Main diagonal is A003024.

Row sums are A053763.

The unlabeled version is A361582.

Cf. A189898 (weak components), A361269 (loops allowed), A361591.

Programs

PARI

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) )}
T(n)={my(e=2); [Vecrev(p) | p<-Vec(serlaplace(U(e, 1/G(e, exp(y*log(U(e, 1/G(e, DigraphEgf(n, e)))))))))]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) }

Formula

T(n,k) = A361269(n,k)/2^n.