A290763 Number of minimal edge covers in the n-sun graph.

17, 56, 207, 839, 3579, 16124, 76037, 373772, 1907842, 10080307, 54988156, 308997810, 1785241070, 10586718392, 64343528516, 400271482199, 2545649131486, 16533901290930, 109563921896553, 740108482190948, 5092272608657314, 35661352536071043

Offset: 3

Author

Eric W. Weisstein, Aug 10 2017

Keywords

• nonn

Links

• Andrew Howroyd, Minimal edge covers in the n-sun graph
• Eric Weisstein's World of Mathematics, Minimal Edge Cover
• Eric Weisstein's World of Mathematics, Sun Graph

Crossrefs

Cf. A053530, A290933.

Programs

Mathematica

b[n_] := b[n] = n!*SeriesCoefficient[Exp[-x-x^2/2 + x*Exp[x]], {x, 0, n}];
a[n_] := Sum[b[i]*Sum[Binomial[j, i]*(2*Binomial[n, 2*j]*(n - j)^(j - i) + Sum[n*Binomial[j + k - 1, j]*Binomial[n - k - 1, 2*k + 2*j - 1]*(n - 2*k - j)^(j - i)/k, {k, 1, (n - 2*j)/3}]), {j, i, n/2}], {i, 0, n/2}];
Table[a[n], {n, 3, 24}] (* Jean-François Alcover, Oct 02 2017, after Andrew Howroyd *)

PARI

\\ here b(n) is A053530
b(n)={Vec(serlaplace(exp(-x-1/2*x^2+x*exp(x + O(x^(n+1))))))[n+1]}
a(n) ={sum(i=0, n\2, b(i)*sum(j=i, n\2, binomial(j,i)*(2*binomial(n,2*j)*(n-j)^(j-i) + sum(k=1, (n-2*j)\3, n*binomial(j+k-1,j)*binomial(n-k-1,2*k+2*j-1)*(n-2*k-j)^(j-i)/k) )))} \\ Andrew Howroyd, Aug 13 2017

Formula

a(n) = Sum_{i=0..n/2} Sum_{j=i..n/2} binomial(j,i)*A053530(i)*(2*binomial(n,2*j)*(n-j)^(j-i) + Sum_{k=1..(n-2*j)/3} n*binomial(j+k-1,j)*binomial(n-k-1,2*k+2*j-1)*(n-2*k-j)^(j-i)/k). - Andrew Howroyd, Aug 13 2017

Extensions

a(6)-a(9) from Andrew Howroyd, Aug 11 2017

a(10)-a(24) from Andrew Howroyd, Aug 13 2017