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

Author

N. J. A. Sloane, Mar 27 2012

Keywords

• nonn

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

Crossrefs

Cf. A053530 (complete graph), A210654.

Programs

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 *)

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

Extensions

More terms from Alois P. Heinz, Feb 10 2013