A290715 - OEIS

%I #20 Aug 10 2017 05:19:18

%S 12,82,1540,35786,880372,30032066,1234252432,57364282990,

%T 3118120533196,194664165928178,13642997281164016,1068856625530082390,

%U 93052682387512347676,8925752446376598352186,937682295833817289298944,107371680361648855572333662

%N Number of minimal edge covers in the n-barbell graph.

%H Andrew Howroyd, <a href="/A290715/b290715.txt">Table of n, a(n) for n = 3..100</a>

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

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

%F a(n) = A053530(n)^2 + A053530(n-1)*(A053530(n-1) + 2 + 2*Sum{i=1..n-2} binomial(n-1,i)*A053530(i)). - _Andrew Howroyd_, Aug 10 2017

%t b[n_] := n! Sum[1/k! (Binomial[k, n - k] 2^(k - n) (-1)^k + Sum[Binomial[k, j] Sum[j^(i - j)/(i - j)! Binomial[k - j, n - i - k + j] 2^(i - j + k - n) (-1)^(k - j), {i, j, n - k + j}], {j, k}]), {k, n}]; Table[b[n]^2 + b[n - 1] (b[n - 1] + 2 + 2 Sum[Binomial[n - 1, i] b[i], {i, n - 2}]), {n, 3, 20}] (* _Eric W. Weisstein_, Aug 10 2017 *)

%o (PARI) \\ here b(n) is A053530

%o b(n)={n!*sum(k=1, n, (binomial(k, n-k)*2^(k-n)*(-1)^k + sum(j=1, k, binomial(k, j) *sum(i=j, n-k+j, j^(i-j)/(i-j)!*binomial(k-j, n-i-k+j)*(1/2)^(n-i-k+j)*(-1)^(k-j))))/k!)}

%o a(n)={my(v=vector(n,i,b(i)));if(n<3,0,v[n]*v[n]+v[n-1]*(v[n-1]+2+2*sum(i=1,n-2,binomial(n-1,i)*v[i])))} \\ _Andrew Howroyd_, Aug 10 2017

%Y Cf. A053530.

%K nonn

%O 3,1

%A _Eric W. Weisstein_, Aug 09 2017

%E a(6)-a(8) from _Giovanni Resta_, Aug 09 2017

%E Terms a(9) and beyond from _Andrew Howroyd_, Aug 10 2017