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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A059027 Number of Dyck paths of semilength n with no peak at height 4. 3
1, 1, 2, 5, 13, 35, 97, 276, 805, 2404, 7343, 22916, 72980, 236857, 782275, 2625265, 8938718, 30834165, 107608097, 379454447, 1350434278, 4845475311, 17512579630, 63703732426, 233063976059, 857067469749, 3166309373615, 11745982220846 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

Peart and Woan, in press, G_4(x).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

P. Peart and W.-J. Woan, Dyck Paths With No Peaks at Height k, J. Integer Sequences, 4 (2001), #01.1.3.

FORMULA

G.f.: (2-3*x+x*(1-4*x)^(1/2))/(2-5*x+x*(1-4*x)^(1/2)).

a(n) = sum(k=1..n-2, sum(j=max(2*k-n+1,0)..k-1, (binomial(k,j)*((k-j)*binomial(2*n-3*k+j-3,n-1-2*k+j)))/(n-k-1)*2^j))+2^(n-1). - Vladimir Kruchinin, Oct 03 2013

a(n) ~ 4^n/(9*sqrt(Pi)*n^(3/2)) * (1+197/(24*n)). - Vaclav Kotesovec, Mar 20 2014

EXAMPLE

1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + 97*x^6 + ...

MATHEMATICA

CoefficientList[Series[(2 - 3 x + x (1 - 4 x)^(1/2))/(2 - 5 x + x (1 - 4 x)^(1/2)), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 05 2013 *)

PROG

a(n):=if n=0 then 1 else sum(sum((binomial(k, j)*((k-j)*binomial(2*n-3*k+j-3, n-1-2*k+j)))/(n-k-1)*2^j, j, max(2*k-n+1, 0), k-1), k, 1, n-2)+2^(n-1); [Vladimir Kruchinin, Oct 03 2013]

(PARI) x='x+O('x^66); Vec((2-3*x+x*(1-4*x)^(1/2))/(2-5*x+x*(1-4*x)^(1/2))) \\ Joerg Arndt, Oct 03 2013

CROSSREFS

G_1, G_2, G_3, G_4 give A000957, A000108, A059019, A059027 resp.

Sequence in context: A085281 A082582 A086581 * A025198 A221205 A037247

Adjacent sequences:  A059024 A059025 A059026 * A059028 A059029 A059030

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 12 2001

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 April 19 08:34 EDT 2019. Contains 322241 sequences. (Running on oeis4.)