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A143370
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Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the grid P_2 x P_n (1<=k<=n). P_m is the path graph on m vertices.
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1
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1, 4, 2, 7, 6, 2, 10, 10, 6, 2, 13, 14, 10, 6, 2, 16, 18, 14, 10, 6, 2, 19, 22, 18, 14, 10, 6, 2, 22, 26, 22, 18, 14, 10, 6, 2, 25, 30, 26, 22, 18, 14, 10, 6, 2, 28, 34, 30, 26, 22, 18, 14, 10, 6, 2, 31, 38, 34, 30, 26, 22, 18, 14, 10, 6, 2, 34, 42, 38, 34, 30, 26, 22, 18, 14, 10, 6, 2
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Sum of entries in row n = n(2n-1)=A000384(n).
The entries in row n are the coefficients of the Wiener polynomial of the grid P_2 x P_n.
Sum(k*T(n,k),k=1..n)=A131423(n) = the Wiener index of the grid P_2 x P_n.
The average of all distances in the grid P_2 x P_n is (n+2)/3.
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REFERENCES
| B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
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FORMULA
| G.f.=G(q,z)=qz(1+2z+qz)/[(1-qz)(1-z)^2].
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EXAMPLE
| T(2,1)=4 because in the graph P_2 x P_2 (a square) we have 4 distances equal to 1.
Triangle starts:
1;
4,2;
7,6,2;
10,10,6,2;
13,14,10,6,2;
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MAPLE
| G:=q*z*(1+2*z+q*z)/((1-z)^2*(1-q*z)): Gser:= simplify(series(G, z=0, 15)): for n to 12 do p[n]:=sort(coeff(Gser, z, n)) end do: for n to 12 do seq(coeff(p[n], q, j), j=1..n) end do; # yields sequence in triangular form
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CROSSREFS
| A000384.
Sequence in context: A194075 A002560 A124908 * A016695 A125271 A092314
Adjacent sequences: A143367 A143368 A143369 * A143371 A143372 A143373
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 05 2008
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