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

0,2

Comments

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

Links

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

Formula

Inverse Euler transform of A049312.

Example

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.

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 *)

Crossrefs

Cf. A005142, A049312, A123549, A318869.

Sequence in context: A063894 A268619 A024500 * A000087 A145667 A095067

Adjacent sequences: A318867 A318868 A318869 * A318871 A318872 A318873

Keyword

nonn

Author

Andrew Howroyd, Sep 04 2018

Status

approved