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A143939
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Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the cycle C_n (1<=k<=floor(n/2)).
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0
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1, 3, 4, 2, 5, 5, 6, 6, 3, 7, 7, 7, 8, 8, 8, 4, 9, 9, 9, 9, 10, 10, 10, 10, 5, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 6, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 7, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 8, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,2
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COMMENTS
| Row n contains floor(n/2) entries.
The entries in row n are the coefficients of the Wiener polynomial of the cycle C_n.
Sum of entries in row n = n(n-1)/2=A000217(n-1).
Sum(k*T(n,k), k=1..floor(n/2))=the Wiener index of the cycle C_n = A034828(n).
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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
| T(2n+1,k)=2n+1 (1<=k<=n); T(2n,k)=2n (1<=k<=n-1); T(2n,n)=n.
G.f.=G(q,z)=qz^2/(1+z-z^2-qz^3)/[(1-qz^2)^2*(1-z)^2].
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EXAMPLE
| T(4,2)=2 because in C_4 (a square) there are 2 distances equal to 2.
Triangle starts:
1;
3;
4,2;
5,5;
6,6,3;
7,7,7;
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MAPLE
| P:=proc(n) if `mod`(n, 2)=0 then n*(sum(q^j, j=1..(1/2)*n-1))+(1/2)*n*q^((1/2)*n) else n*(sum(q^j, j=1..(1/2)*n-1/2)) end if end proc: for n from 2 to 18 do p[n]:=P(n) end do: for n from 2 to 18 do seq(coeff(p[n], q, j), j=1..floor((1/2)*n)) end do; # yields sequence in triangular form
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CROSSREFS
| A000217, A034828
Sequence in context: A139525 A133570 A117041 * A197269 A201905 A138609
Adjacent sequences: A143936 A143937 A143938 * A143940 A143941 A143942
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 06 2008
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