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A180865
Square array read by antidiagonals: T(m,n) is the Wiener index of the stacked book graph B(m,n) (m>=1, n>=1). B(m,n) is defined as the graph Cartesian product S(m+1) x P(n), where S(m+1) is the star graph on m+1 nodes and P(n) is the path graph on n nodes. The Wiener index of a connected graph is the sum of distances between all unordered pairs of nodes in the graph.
0
1, 4, 8, 9, 25, 25, 16, 52, 72, 56, 25, 89, 145, 154, 105, 36, 136, 244, 304, 280, 176, 49, 193, 369, 506, 545, 459, 273, 64, 260, 520, 760, 900, 884, 700, 400, 81, 337, 697, 1066, 1345, 1451, 1337, 1012, 561, 100, 424, 900, 1424, 1880, 2160, 2184, 1920, 1404, 760
OFFSET
1,2
COMMENTS
T(1,n) = A131423(n).
T(2,n) = A180569(n).
LINKS
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
Eric Weisstein's World of Mathematics, Stacked Book Graph.
FORMULA
T(m,n) = (1/6)n[n^2-(m+1)^2+mn(mn+6m+2n)].
The Wiener polynomial p[n](t) of the graph B(m,n) satisfies the recurrence relation p[n] = p[n-1]+mt+(1/2)m(m-1)t^2+[t+mt+2mt^2+m(m-1)t^3]*sum(t^j,j=0..n-2).
EXAMPLE
T(2,1)=4 because B(2,1) reduces to the path graph P(3) which has 2 pairs of nodes at distance 1 and 1 pair at distance 2.
Square array T(m,n) begins:
1, 8, 25, 56, 105, ...
4, 25, 72, 154, 280, ...
9, 52, 145, 304, 545, ...
16, 89, 244, 506, 900, ...
MAPLE
T := proc (m, n) options operator, arrow: (1/6)*n*(n^2-(m+1)^2+m*n*(m*n+2*n+6*m)) end proc: for n to 10 do seq(T(n+1-j, j), j = 1 .. n) end do; # yields sequence in triangular form
CROSSREFS
Sequence in context: A372034 A173743 A035326 * A063907 A261987 A158340
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Sep 28 2010
STATUS
approved