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A119011 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k valleys strictly above the x-axis (0<=k<=n-2; n>=2). A hill in a Dyck path is a peak at level 1. 1
1, 1, 1, 2, 3, 1, 3, 8, 6, 1, 5, 18, 23, 10, 1, 8, 38, 70, 54, 15, 1, 13, 76, 186, 215, 110, 21, 1, 21, 147, 451, 710, 560, 202, 28, 1, 34, 277, 1025, 2065, 2269, 1288, 343, 36, 1, 55, 512, 2220, 5480, 7854, 6321, 2688, 548, 45, 1, 89, 932, 4634, 13574, 24227, 25830 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,4

COMMENTS

Row sums yield the Fine numbers (A000957). T(n,0)=A000045(n-1) (the Fibonacci numbers). T(n,1)=A006478(n). Sum(k*T(n,k),k=0..n-2)=A119012(n)

LINKS

Table of n, a(n) for n=2..62.

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

FORMULA

G.f.: G(t,z)=1/[1-zr(t,z)]-1, where r=r(t,z) is the Narayana function, defined by (1+r)(1+tr)z=r, r(t,0)=0. See Maple program for the explicit form of G(t,z).

EXAMPLE

T(5,2)=6 because we have uud|ud|uuddd, uuudd|ud|udd, uud|uudd|udd, uuud|ud|uddd, uuud|udd|udd and uud|uud|uddd (the valleys above the x-axis are marked with |).

Triangle starts:

1;

1,1;

2,3,1;

3,8,6,1;

5,18,23,10,1;

MAPLE

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

CROSSREFS

Cf. A000957, A000045, A006478, A119012.

Sequence in context: A295380 A093768 A209419 * A300866 A130477 A226513

Adjacent sequences:  A119008 A119009 A119010 * A119012 A119013 A119014

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, May 08 2006

STATUS

approved

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Last modified September 25 22:38 EDT 2020. Contains 337346 sequences. (Running on oeis4.)