A059027 Number of Dyck paths of semilength n with no peak at height 4.

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

Offset: 0

Author

N. J. A. Sloane, Feb 12 2001

Keywords

• nonn

Examples

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

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.

Crossrefs

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

Programs

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 *)

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

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