cp's OEIS Frontend

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

%I A109078 #20 Sep 08 2022 08:45:19
%S A109078 1,0,1,2,4,6,13,22,46,80,166,296,610,1106,2269,4166,8518,15792,32206,
%T A109078 60172,122464,230252,467842,884236,1794196,3406104,6903352,13154948,
%U A109078 26635774,50922986,103020253,197519942,399300166,767502944,1550554582
%N A109078 Number of symmetric Dyck paths of semilength n and having no hills (i.e., no peaks at level 1).
%C A109078 Column 0 of A109077.
%H A109078 G. C. Greubel, <a href="/A109078/b109078.txt">Table of n, a(n) for n = 0..1000</a>
%F A109078 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).
%F A109078 a(n) ~ 2^(n+3/2)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 20 2014
%F A109078 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
%e A109078 a(4)=4 because we have uudduudd, uudududd, uuududdd and uuuudddd, where u=(1,1), d=(1,-1).
%p A109078 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);
%t A109078 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 *)
%o A109078 (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
%o A109078 (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
%o A109078 (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
%Y A109078 Cf. A109077.
%Y A109078 Bisections are A026641 and A072547.
%K A109078 nonn
%O A109078 0,4
%A A109078 _Emeric Deutsch_, Jun 17 2005