0,3

Row n contains 1+floor(3n/2) terms. Row sums yield A033506.

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. (26) and Table V).

G.f.=(1+tz-t^3*z^2)(1-2tz-t^3*z^2)/[1-(1+3t)z-t(1+t)(2+5t)z^2-t^2*(1+2t)(1-t)z^3+t^4*(2+3t+5t^2)z^4-t^6*(1-t)z^5-t^9*z^6]. The row generating polynomials A[n] satisfy A[n]=(1+3t)A[n-1]+t(2+7t+5t^2)A[n-2]+t^2*(1+t-2t^2)A[n-3]-t^4*(2+3t+5t^2)A[n-4]+t^6*(1-t)A[n-5]+t^9*A[n-6].

T(2,2)=11 because in the P_3 X P_ 2 lattice graph with vertex set {O(0,0),A(1,0),B(1,1),C(1,2),D(0,2),E(0,1)} and edge set {OA,EB,DC,OE,ED,AB,BC} we have the following eleven 2-matchings: {OA,EB},{OA,DC},{EB,DC},{OA,ED},{OA,BC},{DC,OE},{DC,AB},{OE,AB},{OE,BC},{ED,AB}, and {ED,BC}.

Triangle starts:

1;

1,2;

1,7,11,3;

1,12,44,56,18;

1,17,102,267,302,123,11;

G:=(1+t*z-t^3*z^2)*(1-2*t*z-t^3*z^2)/(1-(1+3*t)*z-t*(1+t)*(2+5*t)*z^2-t^2*(1+2*t)*(1-t)*z^3+t^4*(2+3*t+5*t^2)*z^4-t^6*(1-t)*z^5-t^9*z^6): Gser:=simplify(series(G, z=0, 11)): P[0]:=1: for n from 1 to 8 do P[n]:=coeff(Gser, z^n) od:for n from 0 to 8 do seq(coeff(t*P[n], t^k), k=1..floor(3*n/2)+1) od; # yields sequence in triangular form

Cf. A033506, A001835.

Adjacent sequences: A100242 A100243 A100244 this_sequence A100246 A100247 A100248

Sequence in context: A032135 A032039 A075118 this_sequence A095137 A141488 A113042

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 28 2004