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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
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
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, 40}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(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
(Magma) R<x>:=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
(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
CROSSREFS
Cf. A109077.
Bisections are A026641 and A072547.
Sequence in context: A087549 A339291 A372643 * A291738 A321228 A033305
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 17 2005
STATUS
approved

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Last modified July 14 03:51 EDT 2024. Contains 374291 sequences. (Running on oeis4.)