A166295 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n and having k UUDUDD's starting at level 0 (0 <= k <= floor(n/3); U=(1,1), D=(1,-1)).

1, 1, 2, 3, 1, 6, 2, 12, 5, 26, 10, 1, 57, 22, 3, 128, 48, 9, 291, 109, 22, 1, 670, 250, 54, 4, 1558, 582, 129, 14, 3655, 1366, 311, 40, 1, 8639, 3232, 750, 109, 5, 20554, 7696, 1818, 284, 20, 49185, 18432, 4419, 730, 65, 1, 118301, 44368, 10776, 1856, 195, 6

Offset: 0

Author

Emeric Deutsch, Oct 29 2009

Keywords

• nonn
• tabf

Comments

Row n has 1 + floor(n/3) terms.

Sum of entries in row n = A004148(n+1) (the secondary structure numbers).

T(n,0) = A166296(n).

Sum_{k=0..floor(n/3)} k*T(n,k) = A166297(n).

Examples

			T(4,1)=2 because we have UDUUDUDD and UUDUDDUD.
Triangle starts:
   1;
   1;
   2;
   3,  1;
   6,  2;
  12,  5;
  26, 10,  1;
		

Crossrefs

Cf. A004148, A166296, A166297.

Programs

Maple

G := 2/(1-z-z^2+2*z^3-2*t*z^3+sqrt(1-2*z-z^2-2*z^3+z^4)): Gser := simplify(series(G, z = 0, 20)): 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, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form

Formula

G.f.: G(t,z) = 1/(1-z-z^2+z^3-t*z^3-z^3*g), where g = 1+zg + z^2*g + z^3*g^2.