A166299 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.

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, 49, 18, 4, 0, 1, 260, 114, 36, 8, 5, 0, 605, 260, 81, 26, 5, 0, 1, 1418, 604, 193, 52, 10, 6, 0, 3350, 1419, 444, 118, 35, 6, 0, 1, 7967, 3350, 1041, 288, 70, 12, 7, 0, 19055, 7966

Offset

0,10

Comments

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

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

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

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

Links

Table of n, a(n) for n=0..73.

Formula

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

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

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

Example

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).
Triangle starts:
1;
0;
0,1;
1,0;
1,0,1;
2,2,0;
5,2,0,1;
10,4,3,0;

Maple

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

Crossrefs

Cf. A004148, A166300, A075125

Sequence in context: A052079 A291483 A181295 * A182508 A213626 A329687

Adjacent sequences: A166296 A166297 A166298 * A166300 A166301 A166302

Keyword

nonn,tabf

Author

Emeric Deutsch, Nov 07 2009

Status

approved