%I
%S 1,1,1,1,2,2,1,4,6,3,1,9,16,12,4,1,23,44,39,20,5,1,65,128,123,76,30,6,
%T 1,197,392,393,268,130,42,7,1,626,1250,1284,928,505,204,56,8,1,2056,
%U 4110,4287,3216,1880,864,301,72,9,1,6918,13834,14583,11224,6885,3438,1379
%N Triangle read by rows: T(n,k) is number of Dyck paths of semilength n with height of second peak equal to k (n>=1; 0<=k<=n1).
%C Row sums are the Catalan numbers (A000108). T(n,0)=1 (paths have only one peak); The g.f. for column k is kz^(k+1)*c^k/(1z), where c=[1sqrt(14z)]/(2z) is the Catalan function. T(n,1)=A014137(n1); T(n,2)=2*A014138(n3); T(n,3)=3*A001453(n2); T(n,4)=4*A114277(n5); Sum(k*T(n,k), k=0..n1)=A112308(n2).
%F G.f.=[(1tzc)^2+tz^2*c]/[(1z)(1tzc)^2]1, where c=[1sqrt(14z)]/(2z) is the Catalan function.
%e T(4,1)=4 because we have UDUDUDUD, UDUDUUDD, UUDDUDUD and UUUDDDUD, where U=(1,1), D=(1,1).
%e Triangle begins:
%e 1;
%e 1,1;
%e 1,2,2;
%e 1,4,6,3;
%e 1,9,16,12,4;
%p G:=((1t*z*c)^2+t*z^2*c)/(1z)/(1t*z*c)^21: c:=(1sqrt(14*z))/2/z: Gser:=simplify(series(G,z=0,15)): for n from 1 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 12 do seq(coeff(t*P[n],t^j),j=1..n) od; # yields sequence in triangular form
%Y Cf. A000108, A014137, A014138, A001453, A114277, A112308.
%K nonn,tabl
%O 1,5
%A _Emeric Deutsch_, Nov 30 2005
