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A094449 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.

%I A094449 #7 Jan 14 2025 06:01:20
%S A094449 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 A094449 85,65,48,28,0,0,64,470,286,214,156,112,64,0,0,128,1616,982,727,517,
%U A094449 364,256,144,0,0,256,5632,3420,2518,1772,1214,832,576,320,0,0,512,19852
%N A094449 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 A094449 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 (A000108).
%F A094449 G.f.: G(t, z) = (1-t*z)*(1-z)/(1-2*t*z+t*z^2-z*(1-z)*(1-t*z)*C), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.
%e A094449 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 A094449 Triangle begins:
%e A094449   [1];
%e A094449   [0, 1];
%e A094449   [0, 0, 2];
%e A094449   [1, 0, 0, 4];
%e A094449   [4, 2, 0, 0, 8];
%e A094449   [13, 8, 5, 0, 0, 16];
%e A094449   [42, 26, 20, 12, 0, 0, 32];
%p A094449 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 A094449 Cf. A082989, A000108.
%K A094449 nonn,tabl
%O A094449 0,6
%A A094449 _Emeric Deutsch_, Jun 04 2004