A100245 Triangle read by rows: T(n,k) is the number of k-matchings in the P_3 X P_n lattice graph.
1, 1, 2, 1, 7, 11, 3, 1, 12, 44, 56, 18, 1, 17, 102, 267, 302, 123, 11, 1, 22, 185, 758, 1597, 1670, 757, 106, 1, 27, 293, 1654, 5256, 9503, 9401, 4603, 908, 41, 1, 32, 426, 3080, 13254, 35004, 56456, 53588, 27688, 6716, 540, 1, 37, 584, 5161, 28191, 99183
Offset: 0
Examples
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;
References
- 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).
Programs
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Maple
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
Formula
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].
Extensions
Keyword tabl changed to tabf by Michel Marcus, Apr 09 2013
Comments