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%I #22 Nov 15 2014 13:37:45
%S 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,
%T 81,69,51,33,12,46,60,102,129,141,141,132,114,93,66,42,15,61,81,141,
%U 183,207,216,213,198,177,147,117,81,51,18,78,105,186,246,285
%N 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).
%C Number of entries in row n is 2*n+2.
%C The entries in row n are the coefficients of the Hosoya polynomial of T_n.
%C TR(n,0) = A028872(n+2) = number of vertices of T_n.
%C TR(n,1) = A140091(n) = number of edges of T_n.
%C sum(j*TR(n,j), j=0..2n+1) = A033544(n) = the Wiener index of T_n.
%C (1/2)*sum(j*(j+1)TR(n,j), j=0..2n+1) = A248094(n) = the hyper-Wiener index of T_n.
%C sum((-1)^j*TR(n,j), j=0..2n+1) = A002061(n). - _Peter Luschny_, Nov 15 2014
%H Q. H. He, J. Z. Gu, S. J. Xu, and W. H. Chan, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match72/n3/match72n3_835-843.pdf">Hosoya polynomials of hexagonal triangles and trapeziums</a>, MATCH, Commun. Math. Comput. Chem. 72, 2014, 835-843.
%F 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.
%e 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).
%e Triangle starts:
%e 1, 0;
%e 6, 6, 6, 3;
%e 13, 15, 21, 21, 15, 6;
%e 22, 27, 42, 48, 45, 36, 24, 9;
%e 33, 42, 69, 84, 87, 81, 69, 51, 33, 12;
%p 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
%Y Cf. A028872, A140091, A033544, A248094.
%K nonn,tabf
%O 0,3
%A _Emeric Deutsch_, Nov 14 2014
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