A180866 Square array read by antidiagonals: T(n,k) is the Wiener index of the firecracker graph F(n,k) (n>=1, k>=2). F(n,k) is the graph obtained by the concatenation of n k-stars by linking one leaf from each. The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

1, 10, 4, 31, 35, 9, 68, 102, 74, 16, 125, 214, 211, 127, 25, 206, 380, 436, 358, 194, 36, 315, 609, 765, 734, 543, 275, 49, 456, 910, 1214, 1280, 1108, 766, 370, 64, 633, 1292, 1799, 2021, 1925, 1558, 1027, 479, 81, 850, 1764, 2536, 2982, 3030, 2700, 2084, 1326

Offset

1,2

Comments

F(3,4)=Y_Y_Y (roughly).

Links

Table of n, a(n) for n=1..53.

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, Firecracker Graph.

Formula

T(n,k)=(1/6)n(6+6k-7k^2+n^2*k^2+12nk^2-18nk).

The Wiener polynomial of the graph F(n,k) is W(n,k,t)=n*w+t[n(1-t)-(1-t^n)]r^2/(1-t)^2, where w=(k-1)t+(1/2)(k-1)(k-2)t^2 and r=1+t+(k-2)t^2.

Example

T(1,3)=4 because the firecracker graph F(1,3) reduces to a path on 3 nodes, where the distances are 1, 1, and 2.
Square array T(n,k) begins:
1,4,9,16,25,36,49, ...
10,35,74,127,194,275,370, ...
31,102,211,358,543,766,1027, ...
68,214,436,734,1108,1558,2084, ...
125,380,765,1280,1925,2700,3605, ...

Maple

T := proc (n, k) options operator, arrow; (1/6)*n*(6+6*k-7*k^2+n^2*k^2+12*n*k^2-18*n*k) end proc: for n to 10 do seq(T(n+2-i, i), i = 2 .. n+1) end do; # yields sequence in triangular form

Crossrefs

Sequence in context: A040094 A094580 A192027 * A232908 A216653 A267180

Adjacent sequences: A180863 A180864 A180865 * A180867 A180868 A180869

Keyword

nonn,tabl

Author

Emeric Deutsch, Sep 29 2010

Status

approved