A100898 Triangle read by rows: T(n,k) is the number of k-matchings of the fan graph on n+1 vertices (i.e., the join of the path graph on n vertices with one extra vertex).

1, 1, 1, 1, 3, 1, 5, 2, 1, 7, 7, 1, 9, 15, 3, 1, 11, 26, 13, 1, 13, 40, 34, 4, 1, 15, 57, 70, 21, 1, 17, 77, 125, 65, 5, 1, 19, 100, 203, 155, 31, 1, 21, 126, 308, 315, 111, 6, 1, 23, 155, 444, 574, 301, 43, 1, 25, 187, 615, 966, 686, 175, 7, 1, 27, 222, 825, 1530, 1386, 532, 57

Offset

0,5

Comments

Row n contains 1 + ceiling(n/2) terms. The row sums yield A029907.

Links

Table of n, a(n) for n=0..70.

Formula

G.f.: (1-z)(1+t*z)/(1 - z - t*z^2)^2.

Example

T(3,2)=2 because in the graph with vertex set {O,A,B,C} and edge set {AB,BC,OA,OB,OC} the 2-matchings are: {OA,BC} and {OC,AB}.
The triangle starts:
  1;
  1,  1;
  1,  3;
  1,  5,  2;
  1,  7,  7;
  1,  9, 15,  3;
  1, 11, 26, 13;

Maple

G:=(1-z)*(1+t*z)/(1-z-t*z^2)^2:Gser:=simplify(series(G, z=0, 18)):P[0]:=1: for n from 1 to 16 do P[n]:=sort(coeff(Gser, z^n)) od:for n from 0 to 15 do seq(coeff(t*P[n], t^k), k=1..1+ceil(n/2)) od; # yields sequence in triangular form

Crossrefs

Cf. A029907.

Sequence in context: A131032 A130323 A130303 * A101350 A199478 A134867

Adjacent sequences: A100895 A100896 A100897 * A100899 A100900 A100901

Keyword

nonn,tabf

Author

Emeric Deutsch, Jan 10 2005

Status

approved