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

3,1

Links

Table of n, a(n) for n=3..24.

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

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

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 *)

Prog

(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

Crossrefs

Cf. A053530, A290933.

Sequence in context: A253424 A309032 A352282 * A253417 A117390 A141841

Adjacent sequences: A290760 A290761 A290762 * A290764 A290765 A290766

Keyword

nonn

Author

Eric W. Weisstein, Aug 10 2017

Extensions

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

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

Status

approved