cp's OEIS Frontend

A121532 Number of double rises at an even level in all nondecreasing Dyck paths of semilength n. A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

%I A121532 #15 Sep 08 2022 08:45:27
%S A121532 0,0,1,6,24,87,290,926,2861,8640,25634,75015,217100,622620,1772097,
%T A121532 5011394,14093980,39448623,109954398,305344314,845165725,2332485420,
%U A121532 6420202246,17629525871,48304680504,132092031672,360557665825
%N A121532 Number of double rises at an even level in all nondecreasing Dyck paths of semilength n. A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.
%H A121532 E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, <a href="http://dx.doi.org/10.1016/S0012-365X(97)82778-1">Nondecreasing Dyck paths and q-Fibonacci numbers</a>, Discrete Math., 170, 1997, 211-217.
%H A121532 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,-5,15,-1,-4,1).
%F A121532 a(n) = Sum_{k>=0} k*A121531(n,k).
%F A121532 a(n) = A054444(n-2) - A121530(n).
%F A121532 G.f.: x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)). [Corrected by _Georg Fischer_, May 24 2019]
%F A121532 a(n) ~ (3-sqrt(5)) * (3+sqrt(5))^n * n / (5 * 2^(n+1)). - _Vaclav Kotesovec_, Mar 20 2014
%F A121532 Equivalently, a(n) ~ phi^(2*n-2) * n / 5, where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 06 2021
%e A121532 a(3)=1 because we have UDUDUD, UDUUDD, UUDDUD, UUDUDD and UU/UDDD, the double rises at an odd level being indicated by a / (U=(1,1), D=(1,-1)).
%p A121532 g:=z^3*(1-3*z^2+2*z^3-z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): gser:=series(g,z=0,35): seq(coeff(gser,z,n),n=1..32);
%t A121532 Rest[CoefficientList[Series[x^3*(1-3*x^2+2*x^3-x^4)/(1+x)/(1-3*x+x^2)^2/(1-x-x^2), {x, 0, 30}], x]] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o A121532 (PARI) my(x='x+O('x^30)); concat([0,0], Vec(x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)))) \\ _G. C. Greubel_, May 24 2019
%o A121532 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!( x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)) )); // _G. C. Greubel_, May 24 2019
%o A121532 (Sage) a=(x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)) ).series(x, 30).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, May 24 2019
%Y A121532 Cf. A121530, A121531, A054444.
%K A121532 nonn
%O A121532 1,4
%A A121532 _Emeric Deutsch_, Aug 05 2006