%I #2 Mar 30 2012 17:36:23

%S 1,0,0,1,1,0,1,0,1,2,2,0,5,2,0,1,10,4,3,0,22,11,3,0,1,50,22,6,4,0,113,

%T 49,18,4,0,1,260,114,36,8,5,0,605,260,81,26,5,0,1,1418,604,193,52,10,

%U 6,0,3350,1419,444,118,35,6,0,1,7967,3350,1041,288,70,12,7,0,19055,7966

%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having k UUDD's starting at level 0.

%C Sum of entries in row n is the secondary structure number A004148(n-1) (n>=2).

%C Number of entries in row n is 1 + floor(n/2).

%C T(n,0)=A166300(n).

%C Sum(k*T(n,k), k>=0)=A075125(n+2).

%F G.f.=G(t,z)=1/(1 + z - zg - tz^2), where g=g(z) satisfies g=1 + zg(g - 1 + z).

%F G.f. of column k is z^{2k}/(1 + z - zg)^{k+1} (k>=0).

%F G(t,z)=2/[1+z+z^2+sqrt((1+z+z^2)(1-3z+z^2)-2tz^2)].

%e T(7,2)=3 because we have (UUDD)(UUDD)UUUDDD, (UUDD)UUUDDD(UUDD), and UUUDDD(UUDD)(UUDD) (the UUDD's starting at level 0 are shown between parentheses).

%e Triangle starts:

%e 1;

%e 0;

%e 0,1;

%e 1,0;

%e 1,0,1;

%e 2,2,0;

%e 5,2,0,1;

%e 10,4,3,0;

%p G := 2/(1+z+z^2+sqrt((1+z+z^2)*(1-3*z+z^2))-2*t*z^2): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

%Y Cf. A004148, A166300, A075125

%K nonn,tabf

%O 0,10

%A _Emeric Deutsch_, Nov 07 2009