A114506 - OEIS

%I #4 Mar 30 2012 17:36:07

%S 1,1,2,4,1,10,4,27,15,79,50,3,240,168,21,750,568,112,2387,1959,504,12,

%T 7711,6850,2115,120,25214,24211,8536,825,83315,86164,33858,4620,55,

%U 277799,308152,133068,23166,715,933596,1106028,520338,108472,6006

%N Triangle read by rows: T(n,k) is number of Dyck paths of semilength n having k ascents of length 3 (0<=k<=floor(n/3)). Also number of ordered trees with n edges that have k vertices of outdegree 3.

%C Row n has 1+floor(n/3) terms. Row sums yield the Catalan numbers (A000108). Column 0 yields A114507. Sum(kT(n,k),k=0..floor(n/3))=binomial(2n-4,n-3) (A001791).

%F G.f. G=G(t, z) satisfies (1-t)z^4*G^4-(1-t)z^3*G^3+zG^2-G+1=0.

%e T(4,1)=4 because we have UDUUUDDD, UUUDDDUD, UUUDUDDD and UUUDDUDD, where U=(1,1), D=(1,-1).

%e Triangle starts:

%e 1;

%e 1;

%e 2;

%e 4,1;

%e 10,4;

%e 27,15;

%e 79,50,3;

%e 240,168,21;

%Y Cf. A000108, A001791, A114507, A102402, A114508.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Dec 03 2005