%I #3 Mar 30 2012 17:35:58
%S 1,0,1,0,0,2,1,0,0,4,4,2,0,0,8,13,8,5,0,0,16,42,26,20,12,0,0,32,139,
%T 85,65,48,28,0,0,64,470,286,214,156,112,64,0,0,128,1616,982,727,517,
%U 364,256,144,0,0,256,5632,3420,2518,1772,1214,832,576,320,0,0,512,19852
%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having sum of pyramid heights equal to k.
%C A pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis. Here U=(1,1) and D(1,-1). This definition differs from the one in A091866. Column k=0 is A082989. Row sums are the Catalan numbers (Q000108).
%F G.f.=G=G(t, z)= (1-tz)(1-z)/[1-2tz+tz^2-z(1-z)(1-t*z)C), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
%e T(3,3)=4 because there are four Dyck paths of semilength 3 having 3 as sum of pyramid heights: (UD)(UUDD),(UUDD)(UD),(UD)(UD)(UD) and (UUUDDD) (the pyramids are shown between parentheses).
%e Triangle begins:
%e [1];[0, 1];[0, 0, 2];[1, 0, 0, 4];[4, 2, 0, 0, 8];[13, 8, 5, 0, 0, 16];[42, 26, 20, 12, 0, 0, 32];
%p C:=(1-sqrt(1-4*z))/2/z: G:=(1-t*z)*(1-z)/(1-2*t*z+t*z^2-z*C*(1-z)*(1-t*z)): Gserz:=simplify(series(G,z=0,16)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(Gserz,z^n)) od: seq([subs(t=0,P[n]),seq(coeff(P[n],t^k),k=1..n)],n=0..14);
%Y Cf. A082989, A000108.
%K nonn,tabl
%O 0,6
%A _Emeric Deutsch_, Jun 04 2004