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%I #20 Oct 02 2017 19:00:27
%S 0,12,396,21672,1918920,250696980,45304472052,10816917169296,
%T 3296928965854032,1248938916843586140,575559130836761023260,
%U 317049200473798671358392,205722831410326997504441496,155295648728262284680608862692,134934407215203512994225979686660
%N Number of (undirected) paths in the n-cocktail party graph.
%H Andrew Howroyd, <a href="/A286038/b286038.txt">Table of n, a(n) for n = 1..50</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphPath.html">Graph Path</a>
%F a(n) = (1/2) * (-2*n - 1 + Sum_{j=0..n} Sum_{k=2*j..2*n} (-1)^j*2^j*(k-j)! * binomial(n,j) * binomial(2*n-2*j,k-2*j) ). - _Andrew Howroyd_, Jun 19 2017
%t a[n_] := (1/2)*(-2n - 1 + Sum[Sum[(-1)^j*2^j*(k - j)!*Binomial[n, j]* Binomial[2n - 2j, k - 2j], {k, 2j, 2n}], {j, 0, n}]);
%t Array[a, 15] (* _Jean-François Alcover_, Oct 02 2017, after _Andrew Howroyd_ *)
%t Table[(Sum[(-2)^k Binomial[n, k] k! HypergeometricU[k + 1, 2 n + 2 - k, 1], {k, 0, n}] - 2 n - 1)/2, {n, 20}] // FunctionExpand (* _Eric W. Weisstein_, Oct 02 2017 *)
%o (PARI)
%o a(n) = (-2*n-1 + sum(j=0,n, sum(k=2*j,2*n, (-1)^j*2^j*(k-j)! * binomial(n,j) * binomial(2*n-2*j, k-2*j))) )/2; \\ _Andrew Howroyd_, Jun 19 2017
%Y Cf. A167987 (cycles), A007060 (Hamiltonian paths), A129348 (Hamiltonian cycles).
%K nonn
%O 1,2
%A _Eric W. Weisstein_, Jun 15 2017
%E Terms a(7) and beyond from _Andrew Howroyd_, Jun 19 2017
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