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

%I A290763 #20 Feb 16 2025 08:33:50
%S A290763 17,56,207,839,3579,16124,76037,373772,1907842,10080307,54988156,
%T A290763 308997810,1785241070,10586718392,64343528516,400271482199,
%U A290763 2545649131486,16533901290930,109563921896553,740108482190948,5092272608657314,35661352536071043
%N A290763 Number of minimal edge covers in the n-sun graph.
%H A290763 Andrew Howroyd, <a href="/A290763/a290763.txt">Minimal edge covers in the n-sun graph</a>
%H A290763 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimalEdgeCover.html">Minimal Edge Cover</a>
%H A290763 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SunGraph.html">Sun Graph</a>
%F A290763 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
%t A290763 b[n_] := b[n] = n!*SeriesCoefficient[Exp[-x-x^2/2 + x*Exp[x]], {x, 0, n}];
%t A290763 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}];
%t A290763 Table[a[n], {n, 3, 24}] (* _Jean-François Alcover_, Oct 02 2017, after _Andrew Howroyd_ *)
%o A290763 (PARI) \\ here b(n) is A053530
%o A290763 b(n)={Vec(serlaplace(exp(-x-1/2*x^2+x*exp(x + O(x^(n+1))))))[n+1]}
%o A290763 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
%Y A290763 Cf. A053530, A290933.
%K A290763 nonn
%O A290763 3,1
%A A290763 _Eric W. Weisstein_, Aug 10 2017
%E A290763 a(6)-a(9) from _Andrew Howroyd_, Aug 11 2017
%E A290763 a(10)-a(24) from _Andrew Howroyd_, Aug 13 2017