cp's OEIS Frontend

A378810 Number of horizontal steps in all peak and valleyless Motzkin meanders of length n.

%I A378810 #10 Dec 09 2024 06:02:53
%S A378810 0,1,4,13,39,110,300,801,2106,5473,14097,36056,91697,232108,585212,
%T A378810 1470557,3684682,9209417,22967446,57167993,142051519,352427720,
%U A378810 873157093,2160579740,5340150100,13185150903,32523933395,80156852042,197391001215,485723767342
%N A378810 Number of horizontal steps in all peak and valleyless Motzkin meanders of length n.
%C A378810 Motzkin meanders are lattice paths starting at (0,0) with steps Up (0,1), Horizontal (1,0), and Down (0,-1) that stay weakly above the x-axis. Peak and valleyless Motzkin meanders avoid UD and DU.
%F A378810 a(n) = Sum_{k=1..n} A378809(n,k)*k.
%e A378810 For n = 3 we have meanders, UUU, UUH, UHU, UHD, HUU, UHH, HHU, HUH, HHH; giving a total of a(3) = 13 H steps.
%o A378810 (PARI)
%o A378810 A088855(n,k) = {binomial(floor((n-1)/2), floor((k-1)/2))*binomial(ceil((n-1)/2),ceil((k-1)/2))}
%o A378810 A_xy(N) = {my(x='x+O('x^N), h = sum(n=0,N, (1/(1-y*x)^(n+1)) * (if(n<1,1,0) + sum(k=1,n, A088855(n,k)*x^(n+k-1)*(y^(k-1)) )) )); h}
%o A378810 P_xy(N) = Pol(A_xy(N), {x})
%o A378810 A_x(N) = {my(px = deriv(P_xy(N),y), y=1); Vecrev(eval(px))}
%o A378810 A_x(20)
%Y A378810 Cf. A005773, A132894, A308435, A378809.
%K A378810 nonn,easy
%O A378810 0,3
%A A378810 _John Tyler Rascoe_, Dec 08 2024