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!)
A191781 Triangle read by rows: T(n,k) is the number of length-n left factors of Dyck paths having length of the first ascent equal to k (1 <= k <= n). 1
1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 6, 6, 4, 2, 1, 1, 10, 10, 7, 4, 2, 1, 1, 20, 20, 14, 8, 4, 2, 1, 1, 35, 35, 25, 15, 8, 4, 2, 1, 1, 70, 70, 50, 30, 16, 8, 4, 2, 1, 1, 126, 126, 91, 56, 31, 16, 8, 4, 2, 1, 1, 252, 252, 182, 112, 62, 32, 16, 8, 4, 2, 1, 1, 462, 462, 336, 210, 119, 63, 32, 16, 8, 4, 2, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

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

T(n,1) = A001405(n-2) (n>=1).

T(n,2) = A001405(n-2) (n>=2).

Sum_{k=1..n} k*T(n,k) = A191782(n).

LINKS

Table of n, a(n) for n=1..91.

FORMULA

G.f.: G(t,z) = (1 - z*c + t*z^3*c^2)/((1 - z*c)*(1 - t*z)*(1 - t*z^2)), where c = (1-sqrt(1 - 4*z^2))/(2*z^2).

EXAMPLE

T(5,2)=3 because we have UUDDU, UUDUD, and UUDUU, where U=(1,1) and D=(1,-1).

Triangle starts:

  1;

  1, 1;

  1, 1, 1;

  2, 2, 1, 1;

  3, 3, 2, 1, 1;

  6, 6, 4, 2, 1, 1;

MAPLE

c := ((1-sqrt(1-4*z^2))*1/2)/z^2: G := (1-z*c+t*z^3*c^2)/((1-z*c)*(1-t*z)*(1-t*z^2*c)): Gser := simplify(series(G, z = 0, 17)): for n to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 13 do seq(coeff(P[n], t, k), k = 1 .. n) end do; # yields sequence in triangular form

CROSSREFS

Cf. A001405, A191782.

Sequence in context: A027358 A332548 A352685 * A155092 A095133 A334572

Adjacent sequences:  A191778 A191779 A191780 * A191782 A191783 A191784

KEYWORD

nonn,tabl

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 29 04:44 EDT 2022. Contains 354122 sequences. (Running on oeis4.)