A301566 a(n) = Sum_{k=1..n-1} k*A088459(n, k).

0, 2, 15, 82, 405, 1891, 8554, 37850, 164985, 710893, 3036726, 12880847, 54331550, 228089538, 953811972, 3975120810, 16519242465, 68474376025, 283211458750, 1169062910873, 4817380232522, 19819870885230, 81429323786460, 334120527783367, 1369374666890230

Offset: 1

Author

Eric W. Weisstein, Mar 23 2018

Keywords

• nonn

Comments

a(n)/binomial(2*n-1,n-1) gives the mean distance of the n-odd graph.

Sum can be given in closed form involving four terms each consisting of a product of binomials and 3F2's.

Links

• Eric Weisstein's World of Mathematics, Mean Distance
• Eric Weisstein's World of Mathematics, Odd Graph

Crossrefs

Cf. A088459, A136328.

Programs

Mathematica

Table[Sum[k Binomial[n, Ceiling[k/2]] Binomial[n - 1, Floor[k/2]], {k, n - 1}], {n, 20}]

PARI

T(n, k) = binomial(n, ceil(k/2))*binomial(n-1, k\2);
a(n) = sum(k=1, n-1, k*T(n,k)); \\ Altug Alkan, Mar 23 2018

Formula

a(n) = 2*A136328(n)/binomial(2*n-1,n-1). - Andrew Howroyd, Mar 24 2018