A290715 Number of minimal edge covers in the n-barbell graph.

12, 82, 1540, 35786, 880372, 30032066, 1234252432, 57364282990, 3118120533196, 194664165928178, 13642997281164016, 1068856625530082390, 93052682387512347676, 8925752446376598352186, 937682295833817289298944, 107371680361648855572333662

Offset

3,1

Links

Andrew Howroyd, Table of n, a(n) for n = 3..100

Eric Weisstein's World of Mathematics, Barbell Graph

Eric Weisstein's World of Mathematics, Minimal Edge Cover

Formula

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

Mathematica

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 *)

Prog

(PARI) \\ here b(n) is A053530
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!)}
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

Crossrefs

Cf. A053530.

Sequence in context: A163020 A164300 A239180 * A175037 A252179 A102105

Adjacent sequences: A290712 A290713 A290714 * A290716 A290717 A290718

Keyword

nonn

Author

Eric W. Weisstein, Aug 09 2017

Extensions

a(6)-a(8) from Giovanni Resta, Aug 09 2017

Terms a(9) and beyond from Andrew Howroyd, Aug 10 2017

Status

approved