The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A102079 Triangle read by rows: T(n,k) is the number of k-matchings in the C_n X P_2 graph (C_n is the cycle graph on n vertices and P_2 is the path graph on 2 vertices). 2
 1, 6, 5, 1, 9, 18, 4, 1, 12, 42, 44, 9, 1, 15, 75, 145, 95, 11, 1, 18, 117, 336, 420, 192, 20, 1, 21, 168, 644, 1225, 1085, 371, 29, 1, 24, 228, 1096, 2834, 3880, 2588, 696, 49, 1, 27, 297, 1719, 5652, 10656, 11097, 5823, 1278, 76, 1, 30, 375, 2540, 10165, 24626, 35645, 29380, 12535, 2310, 125 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS Row n contains n+1 terms. Equivalently, the n-th row gives the coefficients of the matching-generating polynomial of the n-prism graph. - Eric W. Weisstein, Apr 03 2018 LINKS 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. (19) and Table IV). Eric Weisstein's World of Mathematics, Matching-Generating Polynomial Eric Weisstein's World of Mathematics, Prism Graph FORMULA G.f.: -z^2*(5t^4*z^2-1+t^4*z^3+t^5*z^3-6t-5t^2-2tz-7zt^2+zt^3-t^2*z^2)/[(1+tz)(t^3*z^3-tz^2-2tz-z+1)]. The row generating polynomials A[n] satisfy A[n]=(1+t)A[n-1]+2t(1+t)A[n-2]+ t^2*(1-t)A[n-3]-t^4*A[n-4] with A=1+6t+5t^2, A=1+9t+18t^2+4t^3, A=1+12t+42t^2+44t^3+9t^4 and A=1+15t+75t^2+145t^3+95t^4+11t^5. EXAMPLE T(3,3)=4 because in the graph C_3 X P_2 with vertex set {A,B,C,A',B',C'} and edge set {AB,AC,BC, A'B',A'C',B'C',AA',BB',CC'} we have the following 3-matchings: {AA',BB',CC'}, {AA',BC,B'C'}, {BB',AC,A'C'} and {CC',AB,A'B'} (as a matter of fact, these are perfect matchings). Triangle starts: 1, 6, 5; 1, 9, 18, 4; 1, 12, 42, 44, 9; 1, 15, 75, 145, 95, 11; MAPLE G:=-z^2*(5*t^4*z^2-1+z^3*t^4+z^3*t^5-6*t-5*t^2-2*z*t-7*z*t^2+z*t^3-z^2*t^2)/(z*t+1)/(z^3*t^3-z^2*t-2*z*t-z+1) : Gser:=simplify(series(G, z=0, 13)): for n from 2 to 11 do P[n]:=coeff(Gser, z^n) od:for n from 2 to 11 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form MATHEMATICA CoefficientList[LinearRecurrence[{1 + x, 2 x (1 + x), -(-1 + x) x^2, -x^4}, {1 + x, 1 + 6 x + 5 x^2, 1 + 9 x + 18 x^2 + 4 x^3, 1 + 12 x + 42 x^2 + 44 x^3 + 9 x^4}, {2, 10}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *) CoefficientList[CoefficientList[Series[-( -1 - 6 x - 5 x^2 - 2 x z - 7 x^2 z + x^3 z - x^2 z^2 + 5 x^4 z^2 + x^4 z^3 + x^5 z^3)/((1 + x z) (1 - z - 2 x z - x z^2 + x^3 z^3)), {z, 0, 10}], z], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *) CROSSREFS Cf. A102080, A068397. Sequence in context: A242000 A238181 A197517 * A177938 A112282 A098866 Adjacent sequences:  A102076 A102077 A102078 * A102080 A102081 A102082 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Dec 29 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 May 8 18:28 EDT 2021. Contains 343666 sequences. (Running on oeis4.)