This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A100265 Triangle read by rows: T(n,k) is the number of k-matchings in the P_4 X P_n lattice graph. 2
 1, 1, 3, 1, 1, 10, 29, 26, 5, 1, 17, 102, 267, 302, 123, 11, 1, 24, 224, 1044, 2593, 3388, 2150, 552, 36, 1, 31, 395, 2696, 10769, 25835, 36771, 29580, 12181, 2111, 95, 1, 38, 615, 5566, 31106, 111882, 261965, 395184, 372109, 206206, 60730, 7852, 281, 1, 45 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row sums yield A033507. T(n,2n) yields A005178. 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. (46) and Table VI). LINKS FORMULA G.f.= (1 - zt^2)(z^6*t^12 + z^5*t^10 - 2z^5*t^9 - 4z^4*t^8 - 5z^4*t^7 - 3z^4*t^6 - 2z^3*t^6 + 4z^2*t^4 + 11z^2*t^3 + 3z^2*t^2 + zt^2 + 2zt - 1)/( - 1 + z + t^18*z^9 + z^3*t^2 + 4z^3*t^3 - 4z^3*t^4 - 27z^3*t^5 - 15z^3*t^6 + 5z*t + 3zt^2 + 2tz^2 + 13z^2*t^2 + 21z^2*t^3 + 5z^2*t^4 - 2z^7*t^11 - 3z^7*t^12 - 9z^7*t^13 - 9z^7*t^14 - 3z^4*t^4 - 18z^4*t^5 - 41z^4*t^6 - 40z^4*t^7 - 9z^4*t^8 - z^8*t^14 - z^8*t^16 + z^8*t^15 + 3z^5*t^6 + 14z^5*t^7 + 29z^5*t^8 + 24z^5*t^9 + 21z^5*t^10 - z^6*t^8 + 6z^6*t^10 + 19z^6*t^11 + 5z^6*t^12). The row generating polynomials A[n] satisfy A[n] = (5t + 1 + 3t^2)A[n - 1] + (13t^2 + 21t^3 + 5t^4 + 2t)A[n - 2] + ( - 27t^5 - 15t^6 + t^2 - 4t^4 + 4t^3)A[n - 3] + ( - 40t^7 - 9t^8 - 41t^6 - 18t^5 - 3t^4)A[n - 4] + (29t^8 + 21t^10 + 3t^6 + 24t^9 + 14t^7)A[n - 5] + (6t^10 + 5t^12 - t^8 + 19t^11)A[n - 6] + ( - 9t^13 - 2t^11 - 3t^12 - 9t^14)A[n - 7] + ( - t^16 - t^14 + t^15)A[n - 8] + t^18*A[n - 9] EXAMPLE T(2,4)=5 because in the graph P_4 X P_2 with vertices a(0,0), b(0,1), c(0,2), d(0,3),a'(1,0),b'(1,1),c'(1,2),d'(1,3), we have the following 4-matchings {aa',bb',cc',dd'},{aa',bb',cd,c'd'},{ab,a'b',cc',dd'},{ab,a'b',cd,c'd'} and {aa',bc,b'c',dd'} (perfect matchings, of course). Triangle starts: 1; 1, 3, 1; 1, 10, 29, 26, 5; 1, 17, 102, 267, 302, 123, 11; 1, 24, 224, 1044, 2593, 3388, 2150, 552, 36; MAPLE G:= - (1 + 3*z^3*t^4 + 11*z^3*t^5 + 6*z^3*t^6 - 2*z*t - 2*z*t^2 - 3*z^2*t^2 - 9*z^2*t^3 - 3*z^2*t^4 + z^7*t^14 + 3*z^4*t^6 + 5*z^4*t^7 + 2*z^4*t^8 - 3*z^5*t^8 - 3*z^5*t^9 - 5*z^5*t^10 - 2*z^6*t^11)/( - 1 + z + t^18*z^9 + z^3*t^2 + 4*z^3*t^3 - 4*z^3*t^4 - 27*z^3*t^5 - 15*z^3*t^6 + 5*z*t + 3*z*t^2 + 2*z^2*t + 13*z^2*t^2 + 21*z^2*t^3 + 5*z^2*t^4 - 2*z^7*t^11 - 3*z^7*t^12 - 9*z^7*t^13 - 9*z^7*t^14 - 3*z^4*t^4 - 18*z^4*t^5 - 41*z^4*t^6 - 40*z^4*t^7 - 9*z^4*t^8 - z^8*t^14 - z^8*t^16 + z^8*t^15 + 3*z^5*t^6 + 14*z^5*t^7 + 29*z^5*t^8 + 24*z^5*t^9 + 21*z^5*t^10 - z^6*t^8 + 6*z^6*t^10 + 19*z^6*t^11 + 5*z^6*t^12): 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..2*n + 1) od; # yields sequence in triangular form CROSSREFS Cf. A033507, A005178. Sequence in context: A172339 A060540 A087647 * A086766 A078688 A082466 Adjacent sequences:  A100262 A100263 A100264 * A100266 A100267 A100268 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Dec 28 2004 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 22 02:26 EDT 2019. Contains 325210 sequences. (Running on oeis4.)