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A143944
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Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k from each other in the grid P_n X P_n (1 <= k <= 2n-2), where P_n is the path graph on n vertices.
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1
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4, 2, 12, 14, 8, 2, 24, 34, 32, 20, 8, 2, 40, 62, 68, 60, 40, 20, 8, 2, 60, 98, 116, 116, 100, 70, 40, 20, 8, 2, 84, 142, 176, 188, 180, 154, 112, 70, 40, 20, 8, 2, 112, 194, 248, 276, 280, 262, 224, 168, 112, 70, 40, 20, 8, 2, 144, 254, 332, 380, 400, 394, 364, 312, 240
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OFFSET
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2,1
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COMMENTS
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Row n contains 2n-2 entries.
Sum of entries in row n = n^2*(n^2 - 1)/2 = A083374(n).
The entries in row n are the coefficients of the Wiener (Hosoya) polynomial of the grid P_n X P_n.
Sum_{k=1..2n-2} k*T(n,k) = n^3*(n^2 - 1)/3 = A143945(n) = the Wiener index of the grid P_n X P_n.
The average of all distances in the grid P_n X P_n is 2n/3.
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LINKS
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FORMULA
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Generating polynomial of row n is (2q(1-q^n) - n(1-q^2))^2/(2(1-q)^4) - n^2/2.
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EXAMPLE
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T(2,2)=2 because P_2 X P_2 is a square and there are 2 pairs of vertices at distance 2.
Triangle starts:
4, 2;
12, 14, 8, 2;
24, 34, 32, 20, 8, 2;
40, 62, 68, 60, 40, 20, 8, 2;
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MAPLE
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for n from 2 to 10 do Q[n]:=sort(expand(simplify((1/2)*(2*q*(1-q^n)-n*(1-q^2))^2/(1-q)^4-(1/2)*n^2))) end do: for n from 2 to 9 do seq(coeff(Q[n], q, j), j= 1..2*n-2) end do;
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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