A180862 Square array read by antidiagonals: T(m,n) is the Wiener index of the flower graph F(m,n) (m>=2, n>=1). F(m,n) is the graph obtained by joining with an edge a node in the star graph on m nodes to an end-node of the path graph P_n. The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

4, 10, 10, 18, 20, 20, 28, 32, 35, 35, 40, 46, 52, 56, 56, 54, 62, 71, 79, 84, 84, 70, 80, 92, 104, 114, 120, 120, 88, 100, 115, 131, 146, 158, 165, 165, 108, 122, 140, 160, 180, 198, 212, 220, 220, 130, 146, 167, 191, 216, 240, 261, 277, 286, 286, 154, 172, 196, 224

Offset

2,1

Links

Table of n, a(n) for n=2..60.

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, Star Graph.

Formula

T(m,n) = (m-1)^2+(1/6)n(n^2-19)+(1/2)mn(n+5).

G.f.: G(t,s)=Sum(Sum(T(m,n)*t^m*s^n, n=1..infinity),m=2..infinity) = t^2*s(2-t-s)(ts^2-2ts+s^2-2s+2)/[(1-t)^3*(1-s)^4].

The Wiener polynomial of the graph F(m,n) is (m-1)t+(1/2)(m-1)(m-2)t^2+t(t^n-nt+n-1)/(1-t)^2+t[1+t+(m-2)t^2](1-t^n)/(1-t).

Example

Square array T(i,j) begins:
4, 10, 20, 35, 56, 84, ...
10, 20, 35, 56, 84, 120, ...
18, 32, 52, 79, 114, 158, ...
28, 46, 71, 104, 146, 198, ...

Maple

T := proc (m, n) options operator, arrow: (m-1)^2+(1/6)*n*(n^2-19)+(1/2)*m*n*(n+5) end proc; for n from 2 to 12 do seq(T(n+1-i, i), i = 1 .. n-1) end do; # yields sequence in triangular form

Crossrefs

Sequence in context: A111072 A189895 A310333 * A310334 A310335 A352753

Adjacent sequences: A180859 A180860 A180861 * A180863 A180864 A180865

Keyword

nonn,tabl

Author

Emeric Deutsch, Sep 27 2010

Status

approved