A101307 - OEIS

%I #9 Dec 29 2016 02:29:08

%S 1,1,1,3,2,7,6,1,18,18,6,47,59,24,2,129,188,96,16,362,605,369,90,4,

%T 1038,1948,1395,436,45,3022,6305,5164,1981,315,9,8917,20460,18885,

%U 8568,1830,126,26600,66585,68352,35818,9565,1071,21,80098,217186,245497,145796

%N Triangle read by rows: T(n,k) is the number of ordered trees having n edges and k branches of length 2.

%C Row n has 1+floor(n/2) terms (n>=0).

%C Row sums are the Catalan numbers (A000108).

%C Column k=0 yields A101308.

%C T(2n,n) = A001006(n-1) (n>0) (the Motzkin numbers).

%H Emeric Deutsch, <a href="http://dx.doi.org/10.1016/j.disc.2003.10.021">Ordered trees with prescribed root degrees, node degrees and branch lengths</a>, Discrete Math., 282, 2004, 89-94.

%H J. Riordan, <a href="http://dx.doi.org/10.1016/S0097-3165(75)80010-0">Enumeration of plane trees by branches and endpoints</a>, J. Comb. Theory (A) 19, 1975, 214-222.

%F G.f.: G=G(t, z) satisfies G=1+P+PG(G-1), where P= z/(1-z)+(t-1)z^2 (for the explicit form see the Maple program).

%e Triangle begins:

%e    1;

%e    1,  1;

%e    3,  2;

%e    7,  6,  1;

%e   18, 18,  6;

%p G:=(1+t*z^2-z^2+z^3-t*z^3-sqrt((1+t*z^2-z^2+z^3-t*z^3)*(1-4*z+3*z^2-3*t*z^2-3*z^3+3*t*z^3)))/2/z/(1-z+t*z+z^2-t*z^2): Gserz:=simplify(series(G,z=0,16)): for n from 1 to 14 do P[n]:=sort(coeff(Gserz,z^n)) od: for n from 1 to 14 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields the sequence in triangular form

%Y Cf. A000108, A001006, A101308.

%K nonn,tabf

%O 1,4

%A _Emeric Deutsch_, Dec 22 2004