A258122 The multiplicative Wiener index of the cycle graph C_n (n>=3).

1, 4, 32, 1728, 279936, 429981696, 2641807540224, 198135565516800000, 74300837068800000000000, 415989582513831936000000000000, 13974055172471046820331520000000000000, 8285929429609672784320522302259200000000000000, 34392048668455155319241086527782019661824000000000000000, 2908094259133650016606461590346496281704647737999360000000000000000, 1967201733524639238023450985668890257001862763630451357856563200000000000000000

Offset: 3

Author

Emeric Deutsch, Aug 17 2015

Keywords

• nonn

Comments

The multiplicative Wiener index of a connected simple graph G is defined as the product of distances between all pairs of distinct vertices of G.

In the I. Gutman et al. reference, p. 114, the right-hand side of the formula for the multiplicative Wiener index pi(C_n) of C_n (n even) should be replaced by k^k*((k-1)!)^n.

For the Wiener index of C_n see A034828.

Examples

			a(4) = 4 because the distances between vertices are 1,1,1,1,2,and 2.
a(5) = 32 because the distances between vertices are 1,1,1,1,1,2,2,2,2, and 2.
		

Links

• I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., 40, 2000, 113-116.

Crossrefs

Cf. A034828.

Programs

Maple

a := proc(n) if `mod`(n, 2) = 1 then factorial((1/2)*n-1/2)^n else ((1/2)*n)^((1/2)*n)*factorial((1/2)*n-1)^n end if end proc: seq(a(n), n = 3 .. 17);

Formula

a(n) = (k!)^n if n = 2k + 1 is odd (k>=1); a(n) = k^k((k - 1)!)^n if n = 2k is even (k>=2).