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%I #8 May 24 2021 14:29:12
%S 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,
%T 1,17,77,125,65,5,1,19,100,203,155,31,1,21,126,308,315,111,6,1,23,155,
%U 444,574,301,43,1,25,187,615,966,686,175,7,1,27,222,825,1530,1386,532,57
%N 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).
%C Row n contains 1 + ceiling(n/2) terms. The row sums yield A029907.
%F G.f.: (1-z)(1+t*z)/(1 - z - t*z^2)^2.
%e 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}.
%e The triangle starts:
%e 1;
%e 1, 1;
%e 1, 3;
%e 1, 5, 2;
%e 1, 7, 7;
%e 1, 9, 15, 3;
%e 1, 11, 26, 13;
%p 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
%Y Cf. A029907.
%K nonn,tabf
%O 0,5
%A _Emeric Deutsch_, Jan 10 2005
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