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A302719 Number of edge covers in the n-path complement graph.

%I A302719 #19 Apr 12 2025 09:42:29
%S A302719 0,0,0,2,26,580,23116,1703182,237842582,64143512608,33852316389688,
%T A302719 35268292090882874,72930742736413804146,300323342846133370497564,
%U A302719 2467442527810798875863471748,40490661363717159406441954638982,1327931037076594186049396631983031214
%N A302719 Number of edge covers in the n-path complement graph.
%H A302719 Andrew Howroyd, <a href="/A302719/b302719.txt">Table of n, a(n) for n = 1..50</a>
%H A302719 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeCover.html">Edge Cover</a>
%H A302719 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathComplementGraph.html">Path Complement Graph</a>
%F A302719 a(n) = Sum_{i=0..n} (Sum_{k=0..n-i} binomial(n-i, k)*A006129(k)*(2^i)^k*(2^i-1)^(n-i-k)) * (Sum_{j=ceiling(i/2)..i} (-1)^j*binomial(n-j, i-j)*binomial(j-1, 2*j-i)*2^binomial(i, 2)/2^j). - _Andrew Howroyd_, Apr 23 2018
%t A302719 Table[Sum[Sum[Binomial[n - i, k] Sum[(-1)^(k - j) Binomial[k, j] 2^Binomial[j, 2], {j, 0, k}] (2^i)^k If[i == 0 && k == n, 1, (2^i - 1)^(n - i - k)], {k, 0, n - i}] Sum[(-1)^j Binomial[n - j, i - j] Binomial[j - 1, 2 j - i] 2^(Binomial[i, 2] - j), {j, Ceiling[i/2], i}], {i, 0, n}], {n, 10}] (* _Eric W. Weisstein_, Apr 24 2018 *)
%o A302719 (PARI)
%o A302719 a(n)={ my(p=serlaplace(sum(k=0, n, 2^binomial(k,2)*x^k/k!)/exp(x+O(x*x^n))));
%o A302719 sum(i=0, n, sum(k=0, n-i, binomial(n-i,k)*polcoeff(p,k)*(2^i)^k*(2^i-1)^(n-i-k)) * sum(j=(i+1)\2, i, (-1)^j * binomial(n-j, i-j) * binomial(j-1, 2*j-i) * 2^binomial(i,2)/2^j))} \\ _Andrew Howroyd_, Apr 23 2018
%Y A302719 Cf. A006129, A170941.
%K A302719 nonn
%O A302719 1,4
%A A302719 _Eric W. Weisstein_, Apr 12 2018
%E A302719 Terms a(10) and beyond from _Andrew Howroyd_, Apr 23 2018