A380252 Triangular array read by rows: T(n,k) is the number of labeled acyclic digraphs on n vertices with exactly k weakly connected components, n>=0, 0<=k<=n.

1, 0, 1, 0, 2, 1, 0, 18, 6, 1, 0, 446, 84, 12, 1, 0, 26430, 2590, 240, 20, 1, 0, 3596762, 175200, 8970, 540, 30, 1, 0, 1111506858, 26568374, 678930, 24010, 1050, 42, 1, 0, 774460794326, 9127077036, 112393736, 2007600, 54740, 1848, 56, 1, 0, 1206342801843750, 7057099207134, 42191272116, 357391608, 5013540, 111636, 3024, 72, 1

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Formula

E.g.f.: exp(y*log(B(x))) where B(x) = Sum_{n>=0} A003024(n)*x^n/n!.

Example

Triangle T(n,k) begins:
  1;
  0,       1;
  0,       2,      1;
  0,      18,      6,    1;
  0,     446,     84,   12,   1;
  0,   26430,   2590,  240,  20,  1;
  0, 3596762, 175200, 8970, 540, 30, 1;
  ...

Mathematica

nn = 8; B[n_] := n! 2^Binomial[n, 2]; e[x_] := Sum[x^n/B[n], {n, 0, nn}]; egf[ggf_] := Normal[Series[ggf, {x, 0, nn}]] /.Table[x^i -> x^i*2^Binomial[i, 2], {i, 0, nn}]; Table[Drop[(Table[n!, {n, 0, nn}] CoefficientList[Series[Exp[y (Log[egf[1/e[-x]]])], {x, 0, nn}], {x, y}])[[i]], {i + 1, nn + 1}], {i, 1, nn + 1}] // Grid

Crossrefs

Columns k=0-1 give: A000007, A082402.

Row sums give A003024.

Cf. A082403.

Sequence in context: A355565 A202700 A024026 * A355006 A269946 A009829

Adjacent sequences: A380248 A380250 A380251 * A380253 A380254 A380255

Keyword

nonn,tabl,changed

Author

Geoffrey Critzer, Jan 17 2025

Status

approved