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A192022
Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the comb-shaped graph |_|_|...|_| with 2n (n>=1) nodes. The entries in row n are the coefficients of the corresponding Wiener polynomial.
1
1, 0, 3, 2, 1, 5, 5, 4, 1, 7, 8, 8, 4, 1, 9, 11, 12, 8, 4, 1, 11, 14, 16, 12, 8, 4, 1, 13, 17, 20, 16, 12, 8, 4, 1, 15, 20, 24, 20, 16, 12, 8, 4, 1, 17, 23, 28, 24, 20, 16, 12, 8, 4, 1, 19, 26, 32, 28, 24, 20, 16, 12, 8, 4, 1, 21, 29, 36, 32, 28, 24, 20, 16, 12, 8, 4, 1, 23, 32, 40, 36, 32, 28, 24, 20, 16, 12, 8, 4, 1
OFFSET
1,3
COMMENTS
Row n>=2 contains n+1 entries.
Sum of entries in row n is n*(2n-1)=A000384(n) (the hexagonal numbers).
Sum(k*T(n,k),k>=1)=A192023(n) (the Wiener indices).
FORMULA
The generating polynomial of row n (i.e. the Wiener polynomial of the comb with 2n nodes) is n*t + t*(1+t)^2*(n*(1-t)-(1-t^n))/(1-t)^2 or, equivalently, n*t + t*(1+t)^2*Sum((n-j)*t^(j-1),j=1..n-1).
EXAMPLE
T(2,1)=3, T(2,2)=2, T(2,3)=1 because in the graph |_| there are 3 pairs of nodes at distance 1, 2 pairs at distance 2, and 1 pair at distance 3.
Triangle starts:
1,0;
3,2,1;
5,5,4,1;
7,8,8,4,1;
9,11,12,8,4,1;
MAPLE
Q := proc (n) options operator, arrow: n*t+t*(1+t)^2*(sum((n-j)*t^(j-1), j = 1 .. n-1)) end proc: for n to 12 do P[n] := sort(expand(Q(n))) end do: 1; for n from 2 to 12 do seq(coeff(P[n], t, j), j = 1 .. n+1) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 24 2011
STATUS
approved