A101347 Triangle read by rows: T(n,k) is the number of k-matchings of the wheel graph with n spokes.

1, 1, 1, 1, 4, 1, 6, 3, 1, 8, 10, 1, 10, 20, 5, 1, 12, 33, 20, 1, 14, 49, 49, 7, 1, 16, 68, 96, 34, 1, 18, 90, 165, 99, 9, 1, 20, 115, 260, 225, 52, 1, 22, 143, 385, 440, 176, 11, 1, 24, 174, 544, 777, 456, 74, 1, 26, 208, 741, 1274, 1001, 286, 13, 1, 28, 245, 980, 1974, 1960

Offset

0,5

References

C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.

Links

Table of n, a(n) for n=0..68.

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

Row sums yield A061705. T(2n, n)=A005893(n)

Cf. A061705, A005893.

Sequence in context: A153017 A038457 A141649 * A200344 A328726 A127556

Adjacent sequences: A101344 A101345 A101346 * A101348 A101349 A101350

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 25 2004

Status

approved