A109078 Number of symmetric Dyck paths of semilength n and having no hills (i.e., no peaks at level 1).

1, 0, 1, 2, 4, 6, 13, 22, 46, 80, 166, 296, 610, 1106, 2269, 4166, 8518, 15792, 32206, 60172, 122464, 230252, 467842, 884236, 1794196, 3406104, 6903352, 13154948, 26635774, 50922986, 103020253, 197519942, 399300166, 767502944, 1550554582

Offset: 0

Author

Emeric Deutsch, Jun 17 2005

Keywords

• nonn

Comments

Column 0 of A109077.

Examples

			a(4)=4 because we have uudduudd, uudududd, uuududdd and uuuudddd, where u=(1,1), d=(1,-1).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A109077.

Bisections are A026641 and A072547.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 2*(1-x-x*Sqrt(1-4*x^2)+2*x^2 +Sqrt(1-4*x^2))/(1+Sqrt(1-4*x^2)-2*x)/(1+Sqrt(1-4*x^2)+2*x^2) )); // G. C. Greubel, Apr 29 2019

Maple

g:=2*(1-z-z*sqrt(1-4*z^2)+2*z^2+sqrt(1-4*z^2))/(1+sqrt(1-4*z^2)-2*z)/(1+sqrt(1-4*z^2)+2*z^2): gser:=series(g,z=0,39): 1, seq(coeff(gser,z^n),n=1..36);

Mathematica

CoefficientList[Series[2*(1-x-x*Sqrt[1-4*x^2]+2*x^2+Sqrt[1-4*x^2])/(1+ Sqrt[1-4*x^2]-2*x)/(1+Sqrt[1-4*x^2]+2*x^2), {x, 0, 40}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

PARI

my(x='x+O('x^40)); Vec(2*(1-x-x*sqrt(1-4*x^2)+2*x^2 +sqrt(1-4*x^2))/(1+sqrt(1-4*x^2)-2*x)/(1+sqrt(1-4*x^2)+2*x^2)) \\ G. C. Greubel, Mar 16 2017

Sage

(2*(1-x-x*sqrt(1-4*x^2)+2*x^2 +sqrt(1-4*x^2))/(1+sqrt(1-4*x^2)-2*x)/(1+sqrt(1-4*x^2)+2*x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 29 2019

Formula

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

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

D-finite with recurrence 4*(n+1)*a(n) +2*(-n-3)*a(n-1) +2*(-7*n+11)*a(n-2) +(7*n-27)*a(n-3) +2*(-4*n+5)*a(n-4) +4*(n-3)*a(n-5)=0. - R. J. Mathar, Jul 26 2022