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A121485 Number of nondecreasing Dyck paths of semilength n and having no peaks at even level (n>=0). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

%I A121485 #13 Sep 08 2022 08:45:27
%S A121485 1,1,2,4,8,16,33,66,136,274,562,1138,2327,4725,9645,19613,39997,81397,
%T A121485 165906,337773,688260,1401565,2855432,5815477,11846941,24129498,
%U A121485 49152840,100116607,203936639,415394872,846143795,1723513075,3510704795
%N A121485 Number of nondecreasing Dyck paths of semilength n and having no peaks at even level (n>=0). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.
%C A121485 Column 0 of A121484.
%H A121485 Colin Barker, <a href="/A121485/b121485.txt">Table of n, a(n) for n = 1..1000</a>
%H A121485 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 A121485 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-2,-4,0,1).
%F A121485 G.f.: z(1-z^2)(1-2z^2)/(1-z-4z^2+2z^3+4z^4-z^6).
%F A121485 a(n) = a(n-1)+4*a(n-2)-2*a(n-3)-4*a(n-4)+a(n-6) for n>6. - _Colin Barker_, Sep 11 2015
%e A121485 a(4)=4 because we have UDUDUDUD, UDUUUDDD, UUUDDDUD and UUUDUDDD, where U=(1,1) and D=(1,-1).
%p A121485 G:=z*(1-z^2)*(1-2*z^2)/(1-4*z^2-z+4*z^4-z^6+2*z^3): Gser:=series(G,z=0,40): seq(coeff(Gser,z,n),n=1..37);
%t A121485 LinearRecurrence[{1, 4, -2, -4, 0, 1}, {1, 1, 2, 4, 8, 16}, 40] (* _Vincenzo Librandi_, Sep 12 2015 *)
%o A121485 (PARI) Vec(z*(1-z^2)*(1-2*z^2)/(1-z-4*z^2+2*z^3+4*z^4-z^6) + O(z^60)) \\ _Michel Marcus_, Sep 11 2015
%o A121485 (Magma) I:=[1,1,2,4,8,16]; [n le 6 select I[n] else Self(n-1)+4*Self(n-2)-2*Self(n-3)-4*Self(n-4)+Self(n-6): n in [1..40]]; // _Vincenzo Librandi_, Sep 12 2015
%Y A121485 Cf. A121482, A121484.
%K A121485 nonn,easy
%O A121485 1,3
%A A121485 _Emeric Deutsch_, Aug 02 2006