A192024 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the double-comb graph \/\/\/...\/_\/ with 3n (n>=1) nodes. The entries in row n are the coefficients of the corresponding Wiener polynomial.
2, 1, 5, 6, 4, 8, 12, 12, 4, 11, 18, 21, 12, 4, 14, 24, 30, 21, 12, 4, 17, 30, 39, 30, 21, 12, 4, 20, 36, 48, 39, 30, 21, 12, 4, 23, 42, 57, 48, 39, 30, 21, 12, 4, 26, 48, 66, 57, 48, 39, 30, 21, 12, 4, 29, 54, 75, 66, 57, 48, 39, 30, 21, 12, 4, 32, 60, 84, 75, 66, 57, 48, 39, 30, 21, 12, 4
Offset: 1
Examples
T(2,1)=5, T(2,2)=6, T(2,3)=4 because in the graph \/_\/ there are 5 pairs of nodes at distance 1, 6 pairs at distance 2, and 4 pairs at distance 3. Triangle starts: 2,1; 5,6,4; 8,12,12,4; 11,18,21,12,4;
Links
- B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
Programs
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Maple
Q := proc (n) options operator, arrow: n*(t^2+2*t)+t*(1+2*t)^2*(sum((n-j)*t^(j-1), j = 1 .. n-1)) end proc: for n to 11 do P[n] := sort(expand(Q(n))) end do: for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n+1) end do; # yields sequence in triangular form
Formula
G.f.: G(t,z)=t*z*(2+t+z+2*t*z+3*t^2*z)/((1-t*z)*(1-z)^2).
G.f. of column 1: z*(2+z)/(1-z)^2.
G.f. of column 2: z*(1+4*z+z^2)/(1-z)^2.
G.f. of column k>=3: z^(k-1)*(4+4*z+z^2)/(1-z)^2.
The generating polynomial of row n (i.e. the Wiener polynomial of the double-comb with 3n nodes) is n*(2*t +t^2)+t*(1+2*t)^2*(n*(1-t)-(1-t^n))/(1-t)^2 or, equivalently, n*(2*t+t^2)+t*(1+2*t)^2*Sum((n-j)*t^(j-1), j=1..n-1).
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