A121530 - OEIS

%I #11 Jul 26 2022 11:48:51

%S 0,1,4,14,47,148,454,1359,4004,11644,33521,95696,271300,764605,

%T 2143964,5985186,16643779,46124692,127433562,351106955,964976460,

%U 2646158176,7241414949,19779499584,53933402472,146828245753,399137621524

%N Number of double rises at an odd 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.

%C a(n)=Sum(k*A121529(n,k), k>=0). a(n)+A121532(n)=A054444(n-2).

%H 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 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,-5,15,-1,-4,1)

%F G.f.=z^2*(1-2z-z^2+4z^3-3z^4)/[(1+z)(1-3z+z^2)^2*(1-z-z^2)].

%F a(n) ~ (3-sqrt(5)) * (3+sqrt(5))^n * n / (5 * 2^(n+1)). - _Vaclav Kotesovec_, Mar 20 2014

%F Equivalently, a(n) ~ phi^(2*n-2) * n / 5, where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 06 2021

%e a(3)=4 because we have UDUDUD, UDU/UDD, U/UDDUD, U/UDUDD and U/UUDDD, the double rises at an odd level being indicated by a / (U=(1,1), D=(1,-1)).

%p g:=z^2*(1-2*z-z^2+4*z^3-3*z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): gser:=series(g,z=0,33): seq(coeff(gser,z,n),n=1..30);

%t Rest[CoefficientList[Series[x^2*(1-2*x-x^2+4*x^3-3*x^4)/(1+x)/(1-3*x+x^2)^2 /(1-x-x^2), {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Mar 20 2014 *)

%Y Cf. A121529, A121532, A054444.

%K nonn

%O 1,3

%A _Emeric Deutsch_, Aug 05 2006