

A166301


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 pyramid weight k.


1



1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 1, 2, 5, 0, 0, 0, 0, 1, 2, 6, 8, 0, 0, 0, 0, 1, 2, 8, 13, 13, 0, 0, 0, 0, 1, 2, 10, 19, 29, 21, 0, 0, 0, 0, 1, 2, 12, 25, 51, 60, 34, 0, 0, 0, 0, 1, 2, 14, 31, 78, 120, 122, 55, 0, 0, 0, 0, 1, 2, 16, 37, 110, 200, 282, 241
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OFFSET

0,15


COMMENTS

A pyramid in a Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids.
Sum of entries in row n is the secondary structure number A004148(n1) (n>=2).
T(n,n)=A000045(n1) (n>=1; the Fibonacci numbers).
Sum(k*T(n,k), k>=0)=A166302(n).


LINKS

Table of n, a(n) for n=0..89.
A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155176.


FORMULA

G.f.: G=G(t,z) satisfies G = 1 + zG[G  1 + tz  tz(1  t)/(1  tz)].


EXAMPLE

T(6,5)=2 because we have U(UUDD)(UUUDDD)D and U(UUUDDD)(UUDD)D (the maximal pyramids are shown between parentheses).
Triangle starts:
1;
0,0;
0,0,1;
0,0,0,1;
0,0,0,0,2;
0,0,0,0,1,3;
0,0,0,0,1,2,5;
0,0,0,0,1,2,6,8;


MAPLE

eq := G = 1+z*G*(G1+t^2*z*(1z)/(1t*z)): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form


CROSSREFS

Cf. A004148, A166302, A000045, A091866.
Sequence in context: A024157 A039968 A092037 * A187081 A212434 A227186
Adjacent sequences: A166298 A166299 A166300 * A166302 A166303 A166304


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Nov 07 2009


STATUS

approved



