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A121483 Number of peaks at 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.

%I A121483 #8 Jul 26 2022 11:55:41
%S A121483 1,2,6,19,56,167,487,1411,4047,11527,32617,91790,257065,716896,
%T A121483 1991792,5515535,15227846,41930133,115176023,315676425,863475561,
%U A121483 2357539227,6425887551,17487572124,47522431681,128969086382,349567320762
%N A121483 Number of peaks at 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 A121483 a(n) = Sum(k*A121481(n,k),k=0..n).
%H A121483 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 A121483 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,-5,15,-1,-4,1).
%F A121483 G.f.: z(1-z)(1-3z+6z^3-3z^4)/[(1+z)(1-3z+z^2)^2*(1-z-z^2)].
%F A121483 Recurrence: (n^2 - 5*n - 20)*a(n) = (3*n^2 - 12*n - 79)*a(n-1) + (n^2 - 7*n - 16)*a(n-2) - (5*n^2 - 19*n - 138)*a(n-3) - (n^2 - 6*n - 31)*a(n-4) + (n^2 - 3*n - 24)*a(n-5). - _Vaclav Kotesovec_, Mar 20 2014
%F A121483 a(n) ~ (sqrt(5)-1) * (3+sqrt(5))^n * n / (5*2^(n+2)). - _Vaclav Kotesovec_, Mar 20 2014
%e A121483 a(2)=2 because in UDUD and UUDD we have altogether 2 peaks at odd level; here U=(1,1) and D=(1,-1).
%p A121483 G:=z*(1-z)*(1-3*z+6*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 A121483 Rest[CoefficientList[Series[x*(1-x)*(1-3*x+6*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 A121483 Cf. A121481, A121486, A038731.
%K A121483 nonn
%O A121483 1,2
%A A121483 _Emeric Deutsch_, Aug 02 2006