A166284 Number of Dyck paths with no UUU's and no DDD's of semilength n and having k UUDD's (0<=k<=floor(n/2); U=(1,1), D=(1,-1)).

1, 1, 1, 1, 2, 2, 4, 3, 1, 7, 7, 3, 13, 17, 6, 1, 26, 36, 16, 4, 52, 77, 45, 10, 1, 104, 173, 111, 30, 5, 212, 387, 268, 95, 15, 1, 438, 857, 666, 266, 50, 6, 910, 1911, 1641, 714, 175, 21, 1, 1903, 4287, 3975, 1940, 546, 77, 7, 4009, 9619, 9606, 5205, 1610, 294, 28, 1

Offset: 0

Author

Emeric Deutsch, Oct 11 2009

Keywords

• nonn
• tabf

Comments

T(n,k) is also the number of weighted lattice paths B(n) having k (1,0)-steps of weight 2. B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: a (1,0)-step of weight 1; a (1,0)-step of weight 2; a (1,1)-step of weight 2; a (1,-1)-step of weight 1. The weight of a path is the sum of the weights of its steps. Example: row 3 is 2,2; indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the four paths of weight 3 are ud, hH, Hh, and hhh, having 0, 1, 1, and 0 (1,0)-steps of weight 2, respectively. - Emeric Deutsch, Aug 23 2014

Row n contains 1+floor(n/2) entries.

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

T(n,0) = A023431(n).

Sum(k*T(n,k), k=0..floor(n/2)) = A110320(n-1).

Examples

			T(5,2)=3 because we have UDUUDDUUDD, UUDDUDUUDD, and UUDDUUDDUD.
Triangle starts:
1;
1;
1,1;
2,2;
4,3,1;
7,7,3;
13,17,6,1;
26,36,16,4;
		

Links

• M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306. - _Emeric Deutsch_, Aug 23 2014

Crossrefs

Cf. A004148, A023431, A110320.

Programs

Maple

F := RootOf(G = 1+z*G+t*z^2*G+z^3*G^2, G): Fser := series(F, z = 0, 18): for n from 0 to 15 do P[n] := sort(coeff(Fser, z, n)) end do: for n from 0 to 14 do seq(coeff(P[n], t, j), j = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Formula

G.f. G=G(t,z) satisfies G = 1 + zG + tz^2*G + z^3*G^2.