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A378809 Triangle read by rows: T(n,k) is the number of peak and valleyless Motzkin meanders of length n with k horizontal steps.

%I A378809 #14 Mar 31 2025 04:47:16
%S A378809 1,1,1,1,2,1,1,4,3,1,1,5,9,4,1,1,7,15,16,5,1,1,8,27,34,25,6,1,1,10,37,
%T A378809 76,65,36,7,1,1,11,55,124,175,111,49,8,1,1,13,69,216,335,351,175,64,9,
%U A378809 1,1,14,93,309,675,776,637,260,81,10,1
%N A378809 Triangle read by rows: T(n,k) is the number of peak and valleyless Motzkin meanders of length n with k horizontal steps.
%C A378809 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 A378809 G.f.: Sum_{n>=0} 1/(1-y*x)^(n+1) * ([n=0] + Sum_{k=1..n} A088855(n,k)*x^(n+k-1)*y^(k-1)).
%e A378809 The triangle begins
%e A378809    k=0   1   2   3   4   5   6   7
%e A378809  n=0 1;
%e A378809  n=1 1,  1;
%e A378809  n=2 1,  2,  1;
%e A378809  n=3 1,  4,  3,  1;
%e A378809  n=4 1,  5,  9,  4,  1;
%e A378809  n=5 1,  7, 15, 16,  5,  1;
%e A378809  n=6 1,  8, 27, 34, 25,  6,  1;
%e A378809  n=7 1, 10, 37, 76, 65, 36,  7,  1;
%e A378809  ...
%e A378809 T(3,0) = 1: UUU.
%e A378809 T(3,1) = 4: UUH, UHU, UHD, HUU.
%e A378809 T(3,2) = 3: UHH, HHU, HUH.
%e A378809 T(3,3) = 1: HHH.
%o A378809 (PARI)
%o A378809 A088855(n,k) = {binomial(floor((n-1)/2), floor((k-1)/2))*binomial(ceil((n-1)/2),ceil((k-1)/2))}
%o A378809 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)) )) )); for(n=0,N-1,print(Vecrev(polcoeff(h,n))))}
%o A378809 A_xy(10)
%Y A378809 Cf. column k=1 A001651, A005773, A088855, column k=2 A247643, row sums A308435, A378810.
%K A378809 nonn,easy,tabl
%O A378809 0,5
%A A378809 _John Tyler Rascoe_, Dec 08 2024