OFFSET
1,1
COMMENTS
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
TR(m,n) = (4*m^3*(n + 1)^2 + 2*m^2*(3 + 11*n + 6*n^2 - 2*n^3))/3 + ((n*(28 + 45*n - 35*n^2 - 8*n^4)+20*m*(1 + 9*n + 6*n^2 - 4*n^3 + n^4))/30; see Corollary 3,7 in the He et al. reference.
The reader can get the lengthy expression of the bivariate g.f. G by activating the Maple program for TR(m,n) and then activating the Maple program for G.
EXAMPLE
Row 1 is 27; indeed T(1,1) is just one hexagon ABCDEF; it has 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); 6*1 + 6*2 + 3*3 = 27.
Triangle starts:
27;
109, 210;
279, 566, 822;
569, 1182, 1816, 2328;
MAPLE
TR := proc (m, n) options operator, arrow: (4/3)*m^3*(n+1)^2+(2/3)*m^2*(3+11*n+6*n^2-2*n^3)+(1/30)*n*(28+45*n-35*n^2-8*n^4)+(2/3)*m*(1+9*n+6*n^2-4*n^3+n^4) end proc: for m to 10 do seq(TR(m, n), n = 1 .. m) end do; # yields sequence in triangular form
G := factor(sum(sum(TR(i, j)*z^i*t^j, j = 1 .. i), i = 1 .. infinity));
PROG
(Magma) /* As triangle */ [[(4*m^3*(n + 1)^2 + 2*m^2*(3 + 11*n + 6*n^2 - 2*n^3))/3 + ((n*(28 + 45*n - 35*n^2 - 8*n^4)+20*m*(1 + 9*n + 6*n^2 - 4*n^3 + n^4)) / 30): n in [1..m]]: m in [1.. 15]]; // Vincenzo Librandi, Nov 16 2014
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Nov 15 2014
STATUS
approved