2,2

Row 2n contains 3n+1 terms; row 2n+1 contains 3n+2 terms. Row sums yield A102090 T(2n,3n) yields A102091

H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Physics 26 (1985) 157-167 (eq. (51) and Table VII).

The row generating polynomials A[n] satisfy A[n] =(1 + 2t)A[n - 1] + t(3 + 10t + 6t^2)A[n - 2] + t^2*(3 + 7t)A[n - 3] - t^3*( - 1 + 3t + 12t^2 + 10t^3)A[n - 4] - t^5*(3 + 3t + 4t^2)A[n - 5] + t^7*(3 + 2t + 6t^2)A[n - 6] - t^9*(1 - 2t)A[n - 7] - t^12*A[n - 8] G.f.= - z^2*( - 1 - 10t + z^6*t^9 - 3z^5*t^7 - 3z^2*t^2 - 17z^2*t^3 - z^3*t^3 + z^3*t^4 + 3z^4*t^5 + 9z^4*t^6 - 8z^4*t^7 + 33z^3*t^5 - 2z^2*t^4 - 8z^5*t^8 + t^12*z^7 - 4t^8*z^4 + 49t^6*z^3 + 48t^5*z^2 - 3t^9*z^5 - 4t^11*z^6 - 36t^9*z^4 + 40t^7*z^3 + 40t^6*z^2 - 26t^10*z^5 + 2z^7*t^13 + 8t^12*z^6 - 25zt^2 - 47zt^3 - 12zt^4 - 3zt - 24t^2 - 12t^3)/[(z^2*t^3 - 1 - zt)(z^6*t^9 - z^5*t^7 + z^5*t^6 - 5z^4*t^6 - 3z^4*t^5 - 2z^4*t^4 - 2z^3*t^4 + z^3*t^3 + 5z^2*t^3 + z^3*t^2 + 7z^2*t^2 + 2z^2*t + 3zt + z - 1)].

T(2,3)=12 because in the graph C_2 X P_3 with vertex set {A,B,C,A',B',C'} and edge set {AB,AC,A'B',B'C',a,a',b,b',c,c'}, where a and a' are two edges between A and A', b and b' are two edges between B and B', and c and c' are two edges between C and C', we have the following twelve3-matchings (as a matter of fact they are perfect matchings): eight 3-matchings by taking one edge from each of the pairs {a,a'},{b,b'}, and {c,c'}; two 3-matchings by taking AB, A'B' and either edge from the pair {c,c'}; two 3-matchings by taking BC, B'C' and either edge from the pair {a,a'}.

Triangle starts:

1, 10, 24, 12;

1, 15, 69, 107, 36;

1, 20, 142, 440, 588, 288, 32;

1, 25, 240, 1125, 2710, 3227, 1645, 240;

G:= - z^2*( - 1 - 10*t + z^6*t^9 - 3*z^5*t^7 - 3*z^2*t^2 - 17*z^2*t^3 - z^3*t^3 + z^3*t^4 + 3*z^4*t^5 + 9*z^4*t^6 - 8*z^4*t^7 + 33*z^3*t^5 - 2*z^2*t^4 - 8*z^5*t^8 + t^12*z^7 - 4*t^8*z^4 + 49*t^6*z^3 + 48*t^5*z^2 - 3*t^9*z^5 - 4*t^11*z^6 - 36*t^9*z^4 + 40*t^7*z^3 + 40*t^6*z^2 - 26*t^10*z^5 + 2*z^7*t^13 + 8*t^12*z^6 - 25*z*t^2 - 47*z*t^3 - 12*z*t^4 - 3*z*t - 24*t^2 - 12*t^3)/(z^2*t^3 - 1 - z*t)/(z^6*t^9 - z^5*t^7 + z^5*t^6 - 5*z^4*t^6 - 3*z^4*t^5 - 2*z^4*t^4 - 2*z^3*t^4 + z^3*t^3 + 5*z^2*t^3 + z^3*t^2 + 7*z^2*t^2 + 2*z^2*t + 3*z*t + z - 1):

Gser:=simplify(series(G, z=0, 13)): for n from 2 to 9 do P[n]:=coeff(Gser, z^n) od: b:=proc(n) if n mod 2 = 0 then 1 + 3*n/2 else 1 + b(n - 1) fi end:for n from 2 to 9 do seq(coeff(t*P[n], t^k), k=1..b(n)) od; # yields sequence in triangular form

Cf. A102090, A102091.

Sequence in context: A140674 A072245 A135277 this_sequence A133503 A076675 A063925

Adjacent sequences: A102086 A102087 A102088 this_sequence A102090 A102091 A102092

nonn,tabf

Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 29 2004