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
Examples
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;
Crossrefs
Cf. A029907.
Programs
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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
Formula
G.f.: (1-z)(1+t*z)/(1 - z - t*z^2)^2.
Comments