A210655 - OEIS

%I #29 Aug 10 2017 14:53:22

%S 1,2,15,184,2945,63756,1748803,58746304,2361347073,111310111900,

%T 6059192459771,376064819659728,26330615879623393,2061099487899901372,

%U 178985517944285956275,17127853895338704829696,1795558477562697433148417,205139946486547987323752124

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

%C In other words, the number of minimal edge covers in the complete bipartite graph K_{n,n}. - _Andrew Howroyd_, Aug 04 2017

%H Alois P. Heinz, <a href="/A210655/b210655.txt">Table of n, a(n) for n = 1..304</a>

%H Ioan Tomescu, <a href="https://doi.org/10.1016/0095-8956(80)90060-X">Some properties of irreducible coverings by cliques of complete multipartite graphs</a>, J. Combin. Theory Ser. B 28 (1980), no. 2, 127--141. MR0572469 (81i:05106).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MinimalEdgeCover.html">Minimal Edge Cover</a>

%F 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

%p T:= proc(p, q) option remember; `if`(p=1 or q=1, 1,

%p          add(binomial(q, r) *T(p-1, q-r), r=2..q-1)

%p       +q*add(binomial(p-1, s) *T(p-s-1, q-1), s=0..p-2))

%p     end:

%p a:= n-> T(n, n):

%p seq(a(n), n=1..20);  # _Alois P. Heinz_, Feb 10 2013

%t 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_ *)

%t 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 *)

%Y Cf. A053530 (complete graph), A210654.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Mar 27 2012

%E More terms from _Alois P. Heinz_, Feb 10 2013