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A118972 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having length of first descent equal to k (1<=k<=n; n>=1). A hill in a Dyck path is a peak at level 1. 3
0, 0, 1, 1, 0, 1, 3, 2, 0, 1, 10, 5, 2, 0, 1, 33, 16, 5, 2, 0, 1, 111, 51, 16, 5, 2, 0, 1, 379, 168, 51, 16, 5, 2, 0, 1, 1312, 565, 168, 51, 16, 5, 2, 0, 1, 4596, 1934, 565, 168, 51, 16, 5, 2, 0, 1, 16266, 6716, 1934, 565, 168, 51, 16, 5, 2, 0, 1, 58082, 23604, 6716, 1934, 565, 168 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Row sums are the Fine numbers (A000957). T(n,1)=A001558(n-3) for n>=3. T(n,k)=A118973(n-k) for n>=k>=2. Sum(k*T(n,k),k=1..n)=A118974(n)

LINKS

Table of n, a(n) for n=1..72.

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

FORMULA

G:=tz^2*CF[C-(1-t)/(1-tz)], where F=[1-sqrt(1-4z)]/[z(3-sqrt(1-4z)] and C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

EXAMPLE

T(5,2)=5 because we have uu(dd)uududd, uu(dd)uuuddd,uuu(dd)uuddd,uuu(dd)ududd and uuuu(dd)uddd, where u=(1,1), d=(1,-1) (the first descents are shown between parentheses).

Triangle starts:

  0;

  0,1;

  1,0,1;

  3,2,0,1;

  10,5,2,0,1;

  33,16,5,2,0,1;

  ...

MAPLE

F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: G:=t*z^2*C*F*(C-(1-t)/(1-t*z)): Gser:=simplify(series(G, z=0, 15)): for n from 1 to 12 do P[n]:=sort(coeff(Gser, z^n)) od: for n from 1 to 12 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form

CROSSREFS

Cf. A000957, A001558, A118973, A118974.

Sequence in context: A291680 A193283 A193277 * A171224 A270741 A212220

Adjacent sequences:  A118969 A118970 A118971 * A118973 A118974 A118975

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, May 08 2006

STATUS

approved

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Last modified May 24 04:25 EDT 2019. Contains 323528 sequences. (Running on oeis4.)