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A100245
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Triangle read by rows: T(n,k) is the number of k-matchings in the P_3 X P_n lattice graph.
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2
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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
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OFFSET
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0,3
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COMMENTS
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Row n contains 1+floor(3n/2) terms. Row sums yield A033506.
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REFERENCES
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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).
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LINKS
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FORMULA
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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].
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EXAMPLE
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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;
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MAPLE
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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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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