login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A108747 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n and having k returns to the x-axis. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)). 2
2, 2, 4, 4, 8, 8, 10, 20, 24, 16, 28, 56, 72, 64, 32, 84, 168, 224, 224, 160, 64, 264, 528, 720, 768, 640, 384, 128, 858, 1716, 2376, 2640, 2400, 1728, 896, 256, 2860, 5720, 8008, 9152, 8800, 7040, 4480, 2048, 512, 9724, 19448, 27456, 32032, 32032, 27456 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Row sums are the central binomial coefficients (A000984). T(n,1)=2C(n-1), where C(j)=binom(2j,j)/(j+1) is the j-th Catalan number (A000108). T(n,n)=2^n.

Triangle T(n,k), 1<=k<=n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . - Philippe Deléham, Jun 29 2005

LINKS

Table of n, a(n) for n=1..51.

FORMULA

T(n,k) = k * 2^k * binomial(2*n-k,n)/(2*n-k) (1<=k<=n).

G.f.: 1/(1-2*t*z*C), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.

T(n,k) = 2^k * A106566(n,k). - Philippe Deléham, Jun 29 2005

EXAMPLE

T(2,2)=4 because we have u(d)u(d), u(d)d(u), d(u)d(u) and d(u)u(d) (return steps to x-axis shown between parentheses).

Triangle begins:

2;

2,4;

4,8,8;

10,20,24,16;

28,56,72,64,32;

MAPLE

T:= (n, k)-> 2^k*k*binomial(2*n-k, n)/(2*n-k): for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

MATHEMATICA

nn=10; c=(1-(1-4x)^(1/2))/(2x); f[list_]:=Select[list, #>0&]; Map[f, Drop[CoefficientList[Series[1/(1-2y x c), {x, 0, nn}], {x, y}], 1]]//Flatten  (* Geoffrey Critzer, Jan 30 2012 *)

CROSSREFS

Cf. A000984, A000108.

Sequence in context: A059867 A046971 A051754 * A116931 A206558 A145810

Adjacent sequences:  A108744 A108745 A108746 * A108748 A108749 A108750

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jun 23 2005

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 16 14:56 EDT 2019. Contains 324152 sequences. (Running on oeis4.)