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A121486 Number of peaks at 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 A121486 #12 Jul 26 2022 11:46:10
%S A121486 0,1,4,13,43,132,400,1184,3461,9999,28634,81383,229860,645731,1805582,
%T A121486 5028189,13952221,38590922,106434540,292792026,803565215,2200694791,
%U A121486 6015268164,16412564173,44708036568,121600924117,330277253560
%N A121486 Number of peaks at 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 A121486 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 A121486 E. Barcucci, R. Pinzani and R. Sprugnoli, <a href="http://dx.doi.org/10.1007/3-540-56610-4_71">Directed column-convex polyominoes by recurrence relations</a>, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
%H A121486 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,-5,15,-1,-4,1).
%F A121486 a(n) = Sum(k*A121484(n,k),k=0..n-1).
%F A121486 G.f.: z^2*(1-z)(1-z-3z^2+3z^3-z^4)/[(1+z)(1-z-z^2)(1-3z+z^2)^2].
%F A121486 a(n) ~ (sqrt(5)-1) * (3+sqrt(5))^n * n / (5 * 2^(n+2)). - _Vaclav Kotesovec_, Mar 20 2014
%F A121486 20*a(n) = -8*(-1)^n +10*(2*A001871(n)-5*A001871(n-1))+5*(4*A000045(n+1)-7*A000045(n))-3*(4*A001906(n+1)+9*A001906(n)). - _R. J. Mathar_, Jul 26 2022
%e A121486 a(3)=4 because in UDUDUD, UDUU|DD, UU|DDUD, UU|DU|DD and UUUDDD we have altogether 4 peaks at even level (shown by a |); here U=(1,1) and D=(1,-1).
%p A121486 G:=z^2*(1-z)*(1-z-3*z^2+3*z^3-z^4)/(1+z)/(1-z-z^2)/(1-3*z+z^2)^2: Gser:=series(G,z=0,33): seq(coeff(Gser,z,n),n=1..30);
%t A121486 Rest[CoefficientList[Series[x^2*(1-x)*(1-x-3*x^2+3*x^3-x^4)/(1+x)/(1-x-x^2)/(1-3*x+x^2)^2, {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%Y A121486 Cf. A121484, A121483, A038731.
%K A121486 nonn
%O A121486 1,3
%A A121486 _Emeric Deutsch_, Aug 02 2006