A192837 Molecular topological indices of the permutation star graphs.

0, 4, 132, 4680, 214080, 12416400, 896132160, 79295610240, 8481591336960, 1081908144172800, 162548813750400000, 28443681284170521600, 5739117489117031219200, 1323378125974080765388800, 345972881092262536240128000, 101817548412839690547916800000

Offset: 1

Author

Eric W. Weisstein, Jul 11 2011

Keywords

• nonn

Comments

The permutation star graph of order n is a vertex transitive graph with n! vertices and degree n-1. The graph can be constructed as the Cayley graph of the permutations of 1..n with the n-1 generators (1 2), (1 3)..(1 n) where (1 k) is the transposition of 1 and k. The number of nodes at distance k from a specified node is given by A007799(n,k). - Andrew Howroyd, May 13 2017

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• Eric Weisstein's World of Mathematics, Molecular Topological Index
• Eric Weisstein's World of Mathematics, Permutation Star Graph

Crossrefs

Cf. A007799.

Programs

Mathematica

a[n_, 0] = 1; a[n_, 1] = n - 1; a[n_, 2] = (n - 1) (n - 2);
a[n_, k_ /; k >= 2] := a[n, k] = (n - 1) a[n - 1, k - 1] + Sum[j a[j, k - 3], {j, n - 2}];
Table[n! (n - 1) (n - 1 + Sum[k a[n, k], {k, Floor[3 (n - 1)/2]}]), {n, 20}]
(* Eric W. Weisstein, Sep 18 2017 *)

Formula

a(n) = n!*(n-1) * (n-1 + Sum_{k=1..floor(3*(n-1)/2)} k*A007799(n, k)). - Andrew Howroyd, May 13 2017

Extensions

a(7)-a(16) from Andrew Howroyd, May 13 2017