A114588 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at even levels (0<=k<=n-1; n>=1). A hill in a Dyck path is a peak at level 1.

0, 0, 1, 1, 0, 1, 1, 3, 1, 1, 3, 6, 6, 2, 1, 7, 17, 18, 11, 3, 1, 17, 48, 58, 40, 18, 4, 1, 43, 134, 186, 150, 76, 27, 5, 1, 110, 380, 590, 540, 325, 130, 38, 6, 1, 286, 1083, 1860, 1915, 1305, 624, 206, 51, 7, 1, 753, 3100, 5844, 6660, 5115, 2772, 1097, 308, 66, 8, 1, 2003

Offset

1,8

Comments

Row n has n terms. Row sums are the Fine numbers (A000957). Column 0 yields A114589. Sum(k*T(n,k), k=0..n-1) yields A114590.

Links

Table of n, a(n) for n=1..67.

Formula

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

Example

T(4,3) = 1 because we have U(UD)(UD)(UD)D, where U=(1,1), D=(1,-1) (the peaks at even levels are shown between parentheses).
Triangle begins:
0;
0,   1;
1,   0,  1;
1,   3,  1,  1;
3,   6,  6,  2,  1;
7,  17, 18, 11,  3, 1;
17, 48, 58, 40, 18, 4, 1;

Maple

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

Crossrefs

Cf. A000957, A114586, A114587, A114589, A114590.

Sequence in context: A133825 A365968 A156710 * A253223 A121745 A252983

Adjacent sequences: A114585 A114586 A114587 * A114589 A114590 A114591

Keyword

nonn,tabl

Author

Emeric Deutsch, Dec 11 2005

Status

approved