A191788 Triangle read by rows: T(n,k) is the number of length n left factors of Dyck paths having k base pyramids. A base pyramid is a factor of the form U^j D^j (j>0), starting at the horizontal axis.Here U=(1,1) and D=(1,-1).

1, 1, 1, 1, 2, 1, 3, 2, 1, 6, 3, 1, 11, 5, 3, 1, 21, 9, 4, 1, 40, 17, 8, 4, 1, 76, 31, 13, 5, 1, 146, 62, 26, 12, 5, 1, 279, 113, 45, 18, 6, 1, 539, 228, 94, 39, 17, 6, 1, 1036, 419, 165, 64, 24, 7, 1, 2011, 845, 348, 140, 57, 23, 7, 1, 3883, 1568, 618, 237, 89, 31, 8, 1, 7566, 3160, 1298, 521, 205, 81, 30, 8, 1

Offset

0,5

Comments

Row n contains 1+floor(n/2) entries.

Sum of entries in row n is binomial(n,floor(n/2))=A001405(n).

T(n,0) = A191789(n).

Sum(k*T(n,k), k>=0) = A191790(n).

Links

Table of n, a(n) for n=0..80.

Formula

G.f.: G(t,z)=(1-z^2)/((1-z*c)*(1-z^2*c+z^4*c-t*z^2)), where c=(1-sqrt(1-4*z^2))/(2*z^2).

Example

T(6,2)=3 because we have (UD)(UD)UU, (UD)(UUDD), and (UUDD)(UD) (the base pyramids are shown between parentheses).
Triangle starts:
1;
1;
1,1;
2,1;
3,2,1;
6,3,1;
11,5,3,1;
21,9,4,1;

Maple

c := ((1-sqrt(1-4*z^2))*1/2)/z^2: G := (1-z^2)/((1-z*c)*(1-z^2*c+z^4*c-t*z^2)): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Crossrefs

Cf. A001405, A191789, A191790

Sequence in context: A305299 A308701 A191528 * A070979 A363272 A054098

Adjacent sequences: A191785 A191786 A191787 * A191789 A191790 A191791

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 18 2011

Status

approved