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A258122
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The multiplicative Wiener index of the cycle graph C_n (n>=3).
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1, 4, 32, 1728, 279936, 429981696, 2641807540224, 198135565516800000, 74300837068800000000000, 415989582513831936000000000000, 13974055172471046820331520000000000000, 8285929429609672784320522302259200000000000000, 34392048668455155319241086527782019661824000000000000000, 2908094259133650016606461590346496281704647737999360000000000000000, 1967201733524639238023450985668890257001862763630451357856563200000000000000000
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OFFSET
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3,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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