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A127541 Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k even-length branches starting at the root (0<=k<=n). 2

%I

%S 1,1,1,1,3,2,8,5,1,24,15,3,75,46,10,1,243,148,34,4,808,489,116,16,1,

%T 2742,1652,402,61,5,9458,5678,1408,228,23,1,33062,19792,4982,847,97,6,

%U 116868,69798,17783,3138,393,31,1,417022,248577,63967,11627,1557,143,7

%N Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k even-length branches starting at the root (0<=k<=n).

%C Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). T(n,0)=A000958(n-1). Sum(k*T(n,k),k=0..floor(n/2))=A127540(n-1).

%F G.f.=(1+z)/(1+z-z*C-tz^2*C), where C =[1-sqrt(1-4z)]/(2z) is the Catalan function.

%e T(2,0)=1 because we have the tree /\.

%e Triangle starts:

%e 1;

%e 1;

%e 1,1;

%e 3,2;

%e 8,5,1;

%e 24,15,3;

%p C:=(1-sqrt(1-4*z))/2/z: G:=(1+z)/(1+z-z*C-t*z^2*C): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

%Y Cf. A000108, A000958, A127538, A127540.

%K nonn,tabf

%O 0,5

%A _Emeric Deutsch_, Mar 01 2007

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Last modified August 23 12:15 EDT 2019. Contains 326222 sequences. (Running on oeis4.)