A228859 Triangular array read by rows. T(n,k) is the number of labeled bipartite graphs on n nodes having exactly k connected components; n>=1, 1<=k<=n.

1, 1, 1, 3, 3, 1, 19, 15, 6, 1, 195, 125, 45, 10, 1, 3031, 1545, 480, 105, 15, 1, 67263, 27307, 7035, 1400, 210, 21, 1, 2086099, 668367, 140098, 24045, 3430, 378, 28, 1, 89224635, 22427001, 3746925, 536214, 68355, 7434, 630, 36, 1

Offset

1,4

Comments

The Bell transform of A001832(n+1) (without column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 21 2016

Links

Table of n, a(n) for n=1..45.

Formula

E.g.f.: sqrt(A(x)^y) where A(x) is the e.g.f. for A047863.

Sum_{k=1..n} T(n,k)*2^k = A047863(n).

Example

1,
1, 1,
3, 3, 1,
19, 15, 6, 1,
195, 125, 45, 10, 1,
3031, 1545, 480, 105, 15, 1,

Mathematica

nn=9; f[x_]:=Sum[Sum[Binomial[n, k]2^(k(n-k)), {k, 0, n}]x^n/n!, {n, 0, nn}]; Map[Select[#, #>0&]&, Drop[Range[0, nn]!CoefficientList[Series[Exp[y Log[f[x]]/2], {x, 0, nn}], {x, y}], 1]]//Grid

Prog

(Sage) # uses[bell_matrix from A264428, A001832]
# Adds 1, 0, 0, 0, ... as column 0 to the triangle.
bell_matrix(lambda n: A001832(n+1), 8) # Peter Luschny, Jan 21 2016

Crossrefs

Row sums are A047864.

Column 1 is A001832.

Cf. A047863.

Sequence in context: A108391 A111840 A174031 * A259876 A276402 A318110

Adjacent sequences: A228856 A228857 A228858 * A228860 A228861 A228862

Keyword

nonn,tabl

Author

Geoffrey Critzer, Sep 05 2013

Status

approved