A301566 - OEIS

%I #20 Feb 16 2025 08:33:53

%S 0,2,15,82,405,1891,8554,37850,164985,710893,3036726,12880847,

%T 54331550,228089538,953811972,3975120810,16519242465,68474376025,

%U 283211458750,1169062910873,4817380232522,19819870885230,81429323786460,334120527783367,1369374666890230

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

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

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MeanDistance.html">Mean Distance</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddGraph.html">Odd Graph</a>

%F a(n) = 2*A136328(n)/binomial(2*n-1,n-1). - _Andrew Howroyd_, Mar 24 2018

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

%o (PARI) T(n, k) = binomial(n, ceil(k/2))*binomial(n-1, k\2);

%o a(n) = sum(k=1, n-1, k*T(n,k)); \\ _Altug Alkan_, Mar 23 2018

%Y Cf. A088459, A136328.

%K nonn

%O 1,2

%A _Eric W. Weisstein_, Mar 23 2018