A286038 Number of (undirected) paths in the n-cocktail party graph.

0, 12, 396, 21672, 1918920, 250696980, 45304472052, 10816917169296, 3296928965854032, 1248938916843586140, 575559130836761023260, 317049200473798671358392, 205722831410326997504441496, 155295648728262284680608862692, 134934407215203512994225979686660

Offset: 1

Author

Eric W. Weisstein, Jun 15 2017

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• Eric Weisstein's World of Mathematics, Cocktail Party Graph
• Eric Weisstein's World of Mathematics, Graph Path

Crossrefs

Cf. A167987 (cycles), A007060 (Hamiltonian paths), A129348 (Hamiltonian cycles).

Programs

Mathematica

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}]);
Array[a, 15] (* Jean-François Alcover, Oct 02 2017, after Andrew Howroyd *)
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 *)

PARI

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

Formula

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

Extensions

Terms a(7) and beyond from Andrew Howroyd, Jun 19 2017