OFFSET
1,3
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, Dec 09 2017
LINKS
Eric Weisstein's World of Mathematics, Harary Index
Eric Weisstein's World of Mathematics, Permutation Star Graph
FORMULA
a(n)/A296057(n) = (n!/2) * Sum_{k=1..floor(3*(n-1)/2)} A007799(n, k)/k. - Andrew Howroyd, Dec 09 2017
MATHEMATICA
A007799[n_, i_] := Sum[Binomial[n - 1, k] Binomial[n - 1 - k, t] StirlingS1[k + 1, i - k + 1 - 2 t] (-1)^(i + 2 - t), {k, 0, Min[n - 1, i + 1]}, {t, Max[0, Ceiling[(i - 2 k)/2]], Min[n - 1 - k, Floor[(i + 1 - k)/2]]}];
Table[n! Sum[A007799[n, k]/k, {k, Floor[3 (n - 1)/2]}]/2, {n, 20}] // Numerator (* Eric W. Weisstein, Dec 09 2017 *)
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Eric W. Weisstein, Dec 07 2017
EXTENSIONS
a(9)-a(17) from Andrew Howroyd, Dec 09 2017
STATUS
approved