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A143376
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Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the cube Q_n of dimension n (1<=k<=n).
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0
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1, 4, 2, 12, 12, 4, 32, 48, 32, 8, 80, 160, 160, 80, 16, 192, 480, 640, 480, 192, 32, 448, 1344, 2240, 2240, 1344, 448, 64, 1024, 3584, 7168, 8960, 7168, 3584, 1024, 128, 2304, 9216, 21504, 32256, 32256, 21504, 9216, 2304, 256, 5120, 23040, 61440, 107520
(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 = 2^(n-1)*(2^n-1)=A006516.
The entries in row n are the coefficients of the Wiener polynomial of the cube Q_n.
Sum(k*T(n,k),k=1..n)=n*4^(n-1)=A002697(n) = the Wiener index of the cube Q_n.
Triangle T(n,k), 1<=k<=n, read by rows given by [1,1,0,0,0,0,0,...]DELTA[1,1,0,0,0,0,0,...] where DELTA is the operator defined in A084938 ; subtriangle of triangle A055372 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 14 2008]
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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(n,k)=2^(n-1)*binom(n,k).
G.f.=G(q,z)=qz/[(1-2z)(1-2z-2zq).
T(n,k)=A055372(n,k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 14 2008]
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EXAMPLE
| T(2,1)=4, T(2,2)=2 because in Q_1 (a square) there are 4 distances equal to 1 and 2 distances equal to 2.
Triangle starts:
1;
4,2;
12,12,4;
32,48,32,8;
80,160,160,80,16;
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MAPLE
| T:=proc(n, k) options operator, arrow: 2^(n-1)*binomial(n, k) end proc: for n to 10 do seq(T(n, k), k=1..n) end do; # yields sequence in triangular form
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CROSSREFS
| A006516, A002697
Sequence in context: A191441 A152664 A167591 * A111667 A019239 A143944
Adjacent sequences: A143373 A143374 A143375 * A143377 A143378 A143379
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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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EXTENSIONS
| Typo corrected by Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 05 2009
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