A114508 - OEIS

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

%S 1,1,2,5,13,1,37,5,111,21,345,84,1104,322,4,3611,1215,36,12016,4555,

%T 225,40548,17028,1210,138414,63636,5940,22,477076,238004,27534,286,

%U 1657956,891268,122850,2366,5802920,3342375,533625,15925,20436910,12552580

%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having k ascents of length 4 (0<=k<=floor(n/4)). Also number of ordered trees with n edges which have k vertices of outdegree 4.

%C Row n has 1+floor(n/4) terms. Row sums yield the Catalan numbers (A000108). Column 0 yields A114509. Sum(kT(n,k),k=0..floor(n/4))=binomial(2n-5,n-4) (A002054).

%F G.f. G=G(t, z) satisfies (1-t)z^5*G^5-(1-t)z^4*G^4+zG^2-G+1=0.

%e T(5,1)=5 because we have UDUUUUDDDD, UUUDDDDUD, UUUUDDDUDD, UUUUDDUDDD and UUUUDUDDDD, where U=(1,1), D=(1,-1).

%e Triangle starts:

%e 1;

%e 1;

%e 2;

%e 5;

%e 13,1;

%e 37,5;

%e 111,21;

%e 345,84;

%e 1104,322,4;

%e 3611,1215,36;

%p Order:=20: Y:=solve(series((Y-Y^2)/(1-(1-t)*Y^4+(1-t)*Y^5),Y)=z,Y): 1; for n from 1 to 17 do seq(coeff(t*coeff(Y,z^(n+1)),t^j),j=1..1+floor(n/4)) od; # yields sequence in triangular form

%Y Cf. A102402, A114506, A000108, A002054, A114509.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Dec 03 2005