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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 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A028872, A140091, A033544, A248094. Sequence in context: A172360 A175288 A153509 * A292091 A080066 A144539 Adjacent sequences:  A248090 A248091 A248092 * A248094 A248095 A248096 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Nov 14 2014 STATUS approved

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Last modified June 16 14:56 EDT 2019. Contains 324152 sequences. (Running on oeis4.)