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

0, 0, 0, 2, 26, 580, 23116, 1703182, 237842582, 64143512608, 33852316389688, 35268292090882874, 72930742736413804146, 300323342846133370497564, 2467442527810798875863471748, 40490661363717159406441954638982, 1327931037076594186049396631983031214

Offset: 1

Author

Eric W. Weisstein, Apr 12 2018

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• Eric Weisstein's World of Mathematics, Edge Cover
• Eric Weisstein's World of Mathematics, Path Complement Graph

Crossrefs

Cf. A006129, A170941.

Programs

Mathematica

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 *)

PARI

a(n)={ my(p=serlaplace(sum(k=0, n, 2^binomial(k,2)*x^k/k!)/exp(x+O(x*x^n))));
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

Formula

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

Extensions

Terms a(10) and beyond from Andrew Howroyd, Apr 23 2018