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

%I A210655 #30 Feb 16 2025 08:33:17
%S A210655 1,2,15,184,2945,63756,1748803,58746304,2361347073,111310111900,
%T A210655 6059192459771,376064819659728,26330615879623393,2061099487899901372,
%U A210655 178985517944285956275,17127853895338704829696,1795558477562697433148417,205139946486547987323752124
%N A210655 Number of irreducible coverings by edges of the complete bipartite graph K_{n,n}; main diagonal of A210654.
%C A210655 In other words, the number of minimal edge covers in the complete bipartite graph K_{n,n}. - _Andrew Howroyd_, Aug 04 2017
%H A210655 Alois P. Heinz, <a href="/A210655/b210655.txt">Table of n, a(n) for n = 1..304</a>
%H A210655 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 A210655 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H A210655 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimalEdgeCover.html">Minimal Edge Cover</a>
%F A210655 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 A210655 T:= proc(p, q) option remember; `if`(p=1 or q=1, 1,
%p A210655          add(binomial(q, r) *T(p-1, q-r), r=2..q-1)
%p A210655       +q*add(binomial(p-1, s) *T(p-s-1, q-1), s=0..p-2))
%p A210655     end:
%p A210655 a:= n-> T(n, n):
%p A210655 seq(a(n), n=1..20);  # _Alois P. Heinz_, Feb 10 2013
%t A210655 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 A210655 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 A210655 Cf. A053530 (complete graph), A210654.
%K A210655 nonn
%O A210655 1,2
%A A210655 _N. J. A. Sloane_, Mar 27 2012
%E A210655 More terms from _Alois P. Heinz_, Feb 10 2013