A100245 - OEIS

%I #9 May 02 2016 04:50:54

%S 1,1,2,1,7,11,3,1,12,44,56,18,1,17,102,267,302,123,11,1,22,185,758,

%T 1597,1670,757,106,1,27,293,1654,5256,9503,9401,4603,908,41,1,32,426,

%U 3080,13254,35004,56456,53588,27688,6716,540,1,37,584,5161,28191,99183

%N Triangle read by rows: T(n,k) is the number of k-matchings in the P_3 X P_n lattice graph.

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

%D 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).

%F 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].

%e 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}.

%e Triangle starts:

%e 1;

%e 1,2;

%e 1,7,11,3;

%e 1,12,44,56,18;

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

%p 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

%Y Cf. A033506, A001835 (bisection diagonal).

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Dec 28 2004

%E Keyword tabl changed to tabf by _Michel Marcus_, Apr 09 2013