A210655 Number of irreducible coverings by edges of the complete bipartite graph K_{n,n}; main diagonal of A210654.

1, 2, 15, 184, 2945, 63756, 1748803, 58746304, 2361347073, 111310111900, 6059192459771, 376064819659728, 26330615879623393, 2061099487899901372, 178985517944285956275, 17127853895338704829696, 1795558477562697433148417, 205139946486547987323752124

Offset

1,2

Comments

In other words, the number of minimal edge covers in the complete bipartite graph K_{n,n}. - Andrew Howroyd, Aug 04 2017

Links

Alois P. Heinz, Table of n, a(n) for n = 1..304

Ioan Tomescu, Some properties of irreducible coverings by cliques of complete multipartite graphs, J. Combin. Theory Ser. B 28 (1980), no. 2, 127--141. MR0572469 (81i:05106).

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Eric Weisstein's World of Mathematics, Minimal Edge Cover

Formula

a(n) = n!^2 [x^n y^n] exp(x*exp(y)+y*exp(x)-x-y-x*y)-1. - Alois P. Heinz, Feb 10 2013

Maple

T:= proc(p, q) option remember; `if`(p=1 or q=1, 1,
         add(binomial(q, r) *T(p-1, q-r), r=2..q-1)
      +q*add(binomial(p-1, s) *T(p-s-1, q-1), s=0..p-2))
    end:
a:= n-> T(n, n):
seq(a(n), n=1..20);  # Alois P. Heinz, Feb 10 2013

Mathematica

T[p_, q_] := T[p, q] = If[p == 1 || q === 1, 1, Sum[Binomial[q, r]*T[p - 1, q - r], {r, 2, q - 1}] + q*Sum[Binomial[p - 1, s]*T[p - s - 1, q - 1], {s, 0, p - 2}]]; a[n_] := T[n, n]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 24 2014, after Alois P. Heinz *)
With[{ser = Series[Exp[x Exp[y] + y Exp[x] - x - y - x y] - 1, {x, 0, 20}, {y, 0, 20}]}, Table[(n!)^2 Coefficient[ser, x^n y^n], {n, 20}]] (* Eric W. Weisstein, Aug 10 2017 *)

Crossrefs

Cf. A053530 (complete graph), A210654.

Sequence in context: A099709 A208409 A196792 * A052857 A053492 A319834

Adjacent sequences: A210652 A210653 A210654 * A210656 A210657 A210658

Keyword

nonn

Author

N. J. A. Sloane, Mar 27 2012

Extensions

More terms from Alois P. Heinz, Feb 10 2013

Status

approved