OFFSET
0,5
REFERENCES
C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
LINKS
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Eric Weisstein's World of Mathematics, Wheel Graph
FORMULA
T(n, k)=nC(n-k, k)/(n-k)+nC(n-k, k-1) (n>1). G.f.=(1+tz^2)(1-z+tz-tz^2)/(1-z-tz^2)^2.
EXAMPLE
T(3,2)=3 because the 2-matchings of the wheel W_3 (triangle ABC with spokes OA,OB,OC) are (AB,OC), (BC, OA) and (CA, OB).
Triangle begins:
1;
1,1;
1,4;
1,6,3;
1,8,10;
1,10,20,5;
1,12,33,20;
Rows as matching-generating polynomials:
1 + x,
1 + 4*x,
1 + 6*x + 3*x^2,
1 + 8*x + 10*x^2,
1 + 10*x + 20*x^2 + 5*x^3,
...
MAPLE
T:=proc(n, k) if n=0 and k=0 then 1 elif n=1 and k=1 then 1 elif k>ceil(n/2) then 0 else n*binomial(n-k, k)/(n-k)+n*binomial(n-k, k-1) fi end: for n from 0 to 15 do seq(T(n, k), k=0..ceil(n/2)) od; # yields sequence in triangular form
MATHEMATICA
CoefficientList[Table[x^(n/2) (n Sqrt[x] Fibonacci[n, 1/Sqrt[x]] + LucasL[n, 1/Sqrt[x]]), {n, 10}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
CoefficientList[LinearRecurrence[{2, -1 + 2 x, -2 x, -x^2}, {2, 1 + x, 1 + 4 x, 1 + 6 x + 3 x^2}, 10], x] (* Eric W. Weisstein, Apr 03 2018 *)
CoefficientList[CoefficientList[Series[(1 + x - z + 2 x z - 3 x z^2 + x^2 z^2 - 2 x^2 z^3)/(-1 + z + x z^2)^2, {z, 0, 10}], z], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 25 2004
STATUS
approved