cp's OEIS Frontend

A297008 Number of edge covers in the complete tripartite graph K_{n,n,n}.

%I A297008 #14 Feb 16 2025 08:33:52
%S A297008 4,2902,117207580,268752741193822,37231937318464496521924,
%T A297008 323097476641999571450657507823382,
%U A297008 178177528846515370073473806783721111760309500,6274803675843247716007930604166972482973014660984656159102
%N A297008 Number of edge covers in the complete tripartite graph K_{n,n,n}.
%H A297008 Andrew Howroyd, <a href="/A297008/b297008.txt">Table of n, a(n) for n = 1..25</a>
%H A297008 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>
%H A297008 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeCover.html">Edge Cover</a>
%t A297008 b[m_, n_] := Sum[(-1)^j*Binomial[m, j]*If[n == 0, 1, (2^(m - j) - 1)^n], {j, 0, m}];
%t A297008 c[n_, s_] := Sum[Binomial[n, k]*Binomial[n, s - k]*b[k, s - k], {k, Max[0, s - n], Min[n, s]}];
%t A297008 a[n_] := Sum[c[n, 2*n - i]*Sum[(-1)^j*Binomial[i, j]*(2^(2*n - j) - 1)^n, {j, 0, i}], {i, 0, 2 n}];
%t A297008 Array[a, 10] (* _Jean-François Alcover_, Dec 27 2017, after _Andrew Howroyd_ *)
%o A297008 (PARI) \\ here b(m,n) is A183109.
%o A297008 b(m, n)={sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n)}
%o A297008 c(n, s)={sum(k=max(0, s-n), min(n, s),binomial(n, k)*binomial(n, s-k)*b(k, s-k))}
%o A297008 a(n)={sum(i=0, 2*n, c(n, 2*n-i)*sum(j=0, i, (-1)^j*binomial(i, j)*(2^(2*n - j) - 1)^n))} \\ _Andrew Howroyd_, Dec 24 2017
%Y A297008 Cf. A048291, A183109.
%K A297008 nonn
%O A297008 1,1
%A A297008 _Eric W. Weisstein_, Dec 23 2017
%E A297008 Terms a(4) and beyond from _Andrew Howroyd_, Dec 24 2017