The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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). 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A305299 A308701 A191528 * A070979 A363272 A054098
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 18 2011
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 21 17:21 EDT 2024. Contains 372738 sequences. (Running on oeis4.)