cp's OEIS Frontend

A114499 Triangle read by rows: number of Dyck paths of semilength n having k 3-bridges of a given shape (0<=k<=floor(n/3)). A 3-bridge is a subpath of the form UUUDDD or UUDUDD starting at level 0.

%I A114499 #4 Mar 30 2012 17:36:07
%S A114499 1,1,2,4,1,12,2,37,5,119,12,1,390,36,3,1307,114,9,4460,376,25,1,15452,
%T A114499 1262,78,4,54207,4310,255,14,192170,14934,863,44,1,687386,52397,2967,
%U A114499 145,5,2477810,185780,10338,492,20,8992007,664631,36424,1712,70,1
%N A114499 Triangle read by rows: number of Dyck paths of semilength n having k 3-bridges of a given shape (0<=k<=floor(n/3)). A 3-bridge is a subpath of the form UUUDDD or UUDUDD starting at level 0.
%C A114499 Row n has 1+floor(n/3) terms. Row sums are the Catalan numbers (A000108). Column 0 is A114500. Sum(kT(n,k),k=0..floor(n/3))=Catalan(n-2) (n>=3; A000108).
%F A114499 G.f.=1/(1+z^3-tz^3-zC), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
%e A114499 T(4,1)=2 because we have UD(UUUDDD) and (UUUDDD)UD (or UD(UUDUDD) and (UUDUDD)UD). The 3-bridges are shown between parentheses.
%e A114499 Triangle starts:
%e A114499 1;
%e A114499 1;
%e A114499 2;
%e A114499 4,1;
%e A114499 12,2;
%e A114499 37,5;
%e A114499 119,12,1;
%e A114499 390,36,3;
%e A114499 1307,114,9;
%p A114499 C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-z*C+z^3-t*z^3): Gser:=simplify(series(G,z=0,20)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 17 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/3)) od; # yields sequence in triangular form
%Y A114499 Cf. A000108, A114500.
%K A114499 nonn,tabf
%O A114499 0,3
%A A114499 _Emeric Deutsch_, Dec 04 2005