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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 (list; table; graph; refs; listen; history; text; internal format)
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(n-1) (n>=2).

T(n,n)=A000045(n-1) (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, 155-176.

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*(G-1+t^2*z*(1-z)/(1-t*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

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Last modified October 21 08:27 EDT 2018. Contains 316405 sequences. (Running on oeis4.)