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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified June 20 12:24 EDT 2021. Contains 345164 sequences. (Running on oeis4.)