A166285 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n having k peak plateaux (0 <= k <= floor(n/3); U=(1,1), D=(1,-1)).

1, 1, 2, 3, 1, 5, 3, 9, 8, 17, 19, 1, 34, 43, 5, 70, 97, 18, 147, 219, 56, 1, 313, 498, 160, 7, 673, 1140, 438, 32, 1459, 2623, 1168, 122, 1, 3185, 6061, 3062, 418, 9, 6995, 14053, 7932, 1342, 50, 15445, 32677, 20360, 4124, 225, 1, 34265, 76171, 51886, 12274, 895

Offset: 0

Author

Emeric Deutsch, Oct 12 2009

Keywords

• nonn
• tabf

Comments

A peak plateau is a run of consecutive peaks that is preceded by an upstep U and followed by a down step D; a peak consists of an upstep followed by a downstep.

Row n contains 1 + floor(n/3) entries.

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

T(n,0) = A166286(n).

Sum_{k=0..floor(n/3)} k*T(n,k) = A166287(n).

Examples

			T(4,1)=3 because we have UD(UUDUDD), (UUDUDD)UD, and (UUDUDUDD) (the peak plateaux are shown between parentheses).
Triangle starts:
   1;
   1;
   2;
   3,  1;
   5,  3;
   9,  8;
  17, 19,  1;
  34, 43,  5;
		

Crossrefs

Cf. A004148, A166286, A166287.

Programs

Maple

F := RootOf(G = 1+z*G+z^2*G+z^3*G*((t-1)/(1-z)+G), G): Fser := series(F, z = 0, 20): for n from 0 to 17 do P[n] := sort(coeff(Fser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form

Formula

G.f. G=G(t,z) satisfies G = 1 + zG + z^2*G + z^3*G[G+(t-1)/(1-z)].