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A109078 Number of symmetric Dyck paths of semilength n and having no hills (i.e., no peaks at level 1). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Column 0 of A109077.

LINKS

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

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

EXAMPLE

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

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, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

PROG

(PARI) x='x+O('x^50); 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

CROSSREFS

Cf. A109077.

Bisections are A026641 and A072547.

Sequence in context: A058598 A278031 A087549 * A291738 A033305 A105543

Adjacent sequences:  A109075 A109076 A109077 * A109079 A109080 A109081

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jun 17 2005

STATUS

approved

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Last modified December 12 21:50 EST 2017. Contains 295954 sequences.