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A191793 Triangle read by rows: T(n,k) is the number of length n left factors of Dyck paths having k UUDD's, where U=(1,1) and D=(1,-1).

%I A191793 #4 Mar 30 2012 17:36:28
%S A191793 1,1,2,3,5,1,8,2,14,6,23,12,41,28,1,69,54,3,124,116,12,212,220,30,383,
%T A191793 453,87,1,662,852,198,4,1200,1712,500,20,2091,3204,1080,60,3799,6338,
%U A191793 2526,206,1,6661,11824,5280,540,5,12122,23136,11772,1560,30,21359,43068,24066,3780,105
%N A191793 Triangle read by rows: T(n,k) is the number of length n left factors of Dyck paths having k UUDD's, where U=(1,1) and D=(1,-1).
%C A191793 Row n contains 1+floor(n/4) entries.
%C A191793 Sum of entries in row n is binomial(n, floor(n/2)) =A001405(n).
%C A191793 T(n,0)=A191794(n).
%C A191793 Sum(k*T(n,k), k>=0)=A100071(n-3).
%F A191793 G.f.: G(t,z)= C/(1-z*C), where C=C(t,z) is given by z^2*C^2-(1+z^4-t*z^4)*C +1=0.
%e A191793 T(9,2)=3 because we have (UUDD)(UUDD)U, (UUDD)U(UUDD), and U(UUDD)(UUDD), where U=(1,1) and D=(1,-1) (the UUDD's are shown between parentheses).
%e A191793 Triangle starts:
%e A191793 1;
%e A191793 1;
%e A191793 2;
%e A191793 3;
%e A191793 5,1;
%e A191793 8,2;
%e A191793 14,6;
%e A191793 23,12;
%e A191793 41,28,1;
%p A191793 eq := z^2*C^2-(1+z^4-t*z^4)*C+1 = 0: C := RootOf(eq, C): G := C/(1-z*C): Gser := simplify(series(G, z = 0, 23)): for n from 0 to 19 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 19 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*n)) end do; # yields sequence in triangular form
%Y A191793 Cf. A001405, A191794
%K A191793 nonn,tabf
%O A191793 0,3
%A A191793 _Emeric Deutsch_, Jun 18 2011