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A114586 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at odd levels (0<=k<=n-2; n>=2). A hill in a Dyck path is a peak at level 1. 3
1, 1, 1, 3, 2, 1, 6, 8, 3, 1, 15, 22, 15, 4, 1, 36, 68, 52, 24, 5, 1, 91, 198, 191, 100, 35, 6, 1, 232, 586, 651, 425, 170, 48, 7, 1, 603, 1718, 2203, 1656, 820, 266, 63, 8, 1, 1585, 5047, 7285, 6299, 3591, 1435, 392, 80, 9, 1, 4213, 14808, 23832, 23164, 15155, 6972, 2338 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,4

COMMENTS

Row sums are the Fine numbers (A000957). Column 0 yield the Riordan numbers (A005043). Sum(k*T(n,k),k=0..n-2)=A114587(n).

LINKS

Table of n, a(n) for n=2..63.

FORMULA

G.f.=G-1, where G=G(t, z) satisfies z(1+t+z)G^2-(1+z+tz)G+1=0.

EXAMPLE

T(5,2)=3 because we have UU(UD)DU(UD)DD, UUDU(UD)(UD)DD and UU(UD)(UD)DUDD, where U=(1,1), D=(1,-1) (the peaks at odd levels are shown between parentheses).

Triangle begins:

1;

1,1;

3,2,1;

6,8,3,1;

15,22,15,4,1;

MAPLE

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

CROSSREFS

Cf. A000957, A005043, A114587, A114588, A100754.

Sequence in context: A257558 A202390 A210858 * A052174 A227790 A181897

Adjacent sequences:  A114583 A114584 A114585 * A114587 A114588 A114589

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Dec 11 2005

STATUS

approved

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Last modified May 11 05:58 EDT 2021. Contains 343784 sequences. (Running on oeis4.)