A284039 -id:A284039 - cp's OEIS frontend

A296057 Denominators of Harary index for the n-permutation star graph.

1, 1, 1, 1, 1, 1, 1, 1, 11, 143, 13, 13, 17, 19, 323, 323, 437, 437, 23, 1, 667, 20677, 899, 31, 899, 33263, 33263, 33263, 47027, 2022161, 65231, 65231, 3065857, 3065857, 3065857, 3065857, 4391633, 4391633, 4391633, 4391633, 6319667, 385499687, 8965109, 8965109

Offset: 1

Andrew Howroyd, Dec 09 2017

Eric Weisstein's World of Mathematics, Harary Index
Eric Weisstein's World of Mathematics, Permutation Star Graph

Cf. A296190 (numerators), A007799, A284039.

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}] // Denominator (* Eric W. Weisstein, Dec 09 2017 *)

A296190 Numerators of Harary index for the n-permutation star graph.

Original entry on oeis.org

0, 1, 10, 123, 2202, 59040, 2287680, 121394000, 92649740400, 105538103163360, 1034297134668000, 134399089883282400, 27076064087538702720, 5451799851068349018240, 19300076847195336557164800, 4599598343095846092562560000, 1682634821690958905899793664000

Offset: 1

Eric W. Weisstein, Dec 07 2017

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

Eric Weisstein's World of Mathematics, Harary Index
Eric Weisstein's World of Mathematics, Permutation Star Graph

Cf. A296057 (denominators), A007799, A284039.

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 *)

a(n)/A296057(n) = (n!/2) * Sum_{k=1..floor(3*(n-1)/2)} A007799(n, k)/k. - Andrew Howroyd, Dec 09 2017

a(9)-a(17) from Andrew Howroyd, Dec 09 2017

cp's OEIS Frontend

A296057 Denominators of Harary index for the n-permutation star graph.

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