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A248093
Triangle read by rows: TR(n,k) is the number of unordered vertex pairs at distance k of the hexagonal triangle T_n, defined in the He et al. reference (1<=k<=2n+1).
1
1, 0, 6, 6, 6, 3, 13, 15, 21, 21, 15, 6, 22, 27, 42, 48, 45, 36, 24, 9, 33, 42, 69, 84, 87, 81, 69, 51, 33, 12, 46, 60, 102, 129, 141, 141, 132, 114, 93, 66, 42, 15, 61, 81, 141, 183, 207, 216, 213, 198, 177, 147, 117, 81, 51, 18, 78, 105, 186, 246, 285
OFFSET
0,3
COMMENTS
Number of entries in row n is 2*n+2.
The entries in row n are the coefficients of the Hosoya polynomial of T_n.
TR(n,0) = A028872(n+2) = number of vertices of T_n.
TR(n,1) = A140091(n) = number of edges of T_n.
sum(j*TR(n,j), j=0..2n+1) = A033544(n) = the Wiener index of T_n.
(1/2)*sum(j*(j+1)TR(n,j), j=0..2n+1) = A248094(n) = the hyper-Wiener index of T_n.
sum((-1)^j*TR(n,j), j=0..2n+1) = A002061(n). - Peter Luschny, Nov 15 2014
LINKS
Q. H. He, J. Z. Gu, S. J. Xu, and W. H. Chan, Hosoya polynomials of hexagonal triangles and trapeziums, MATCH, Commun. Math. Comput. Chem. 72, 2014, 835-843.
FORMULA
G.f.: (1 + (3 + 6*t + 4*t^2 + 3*t^3)*z - (1 + t + 2*t^2)*(2 + t - 2*t^2)*z^2 +t^2*(1 - 3*t^2)*z^3 + t^4*z^4)/((1-z)^3*(1 - t^2*z^2)^2); follows from Theorem 3.6 of the He et al. reference.
EXAMPLE
Row n=1 is 6, 6, 6, 3; indeed, T_1 is a hexagon ABCDEF; it has 6 distances equal to 0 (= number of vertices), 6 distances equal to 1 (= number of edges), 6 distances equal to 2 (AC, BD, CE, DA, EA, FB), and 3 distances equal to 3 (AD, BE, CF).
Triangle starts:
1, 0;
6, 6, 6, 3;
13, 15, 21, 21, 15, 6;
22, 27, 42, 48, 45, 36, 24, 9;
33, 42, 69, 84, 87, 81, 69, 51, 33, 12;
MAPLE
G := (1+(3+6*t+4*t^2+3*t^3)*z-(1+t+2*t^2)*(2+t-2*t^2)*z^2+t^2*(1-3*t^2)*z^3+t^4*z^4)/((1-z)^3*(1-t^2*z)^2): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 22 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j = 0 .. 2*n+1) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Nov 14 2014
STATUS
approved