A318870 Number of connected bipartite graphs on n unlabeled nodes with a distinguished bipartite block.

1, 2, 1, 2, 4, 10, 27, 88, 328, 1460, 7799, 51196, 422521, 4483460, 62330116, 1150504224, 28434624153, 945480850638, 42417674401330, 2572198227615998, 211135833162079184, 23487811567341121158, 3545543330739039981738, 727053904070651775719646

Offset: 0

Author

Andrew Howroyd, Sep 04 2018

Keywords

• nonn

Comments

Essentially the same as A007776. - Georg Fischer, Oct 02 2018

Examples

			a(1) = 2 because the single node can either be in the distinguished bipartite block or not.
a(2) = 1 because the only connected bipartite graph on two nodes is the complete graph on two nodes.
a(3) = 2 because the only connected bipartite graph on three nodes is the path graph on three nodes and there is a choice about which nodes are in the distinguished block.
		

Links

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

Crossrefs

Cf. A005142, A049312, A123549, A318869.

Programs

Mathematica

mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
b[d_] := Sum[A[n, d - n], {n, 0, d}];
Join[{1}, EULERi[Array[b, 23]]] (* Jean-François Alcover, Sep 13 2018, after Alois P. Heinz in A049312 *)

Formula

Inverse Euler transform of A049312.