A378810 -id:A378810 - cp's OEIS frontend

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

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, 76, 65, 36, 7, 1, 1, 11, 55, 124, 175, 111, 49, 8, 1, 1, 13, 69, 216, 335, 351, 175, 64, 9, 1, 1, 14, 93, 309, 675, 776, 637, 260, 81, 10, 1

Offset: 0

John Tyler Rascoe, Dec 08 2024

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.

			The triangle begins
   k=0   1   2   3   4   5   6   7
 n=0 1;
 n=1 1,  1;
 n=2 1,  2,  1;
 n=3 1,  4,  3,  1;
 n=4 1,  5,  9,  4,  1;
 n=5 1,  7, 15, 16,  5,  1;
 n=6 1,  8, 27, 34, 25,  6,  1;
 n=7 1, 10, 37, 76, 65, 36,  7,  1;
 ...
T(3,0) = 1: UUU.
T(3,1) = 4: UUH, UHU, UHD, HUU.
T(3,2) = 3: UHH, HHU, HUH.
T(3,3) = 1: HHH.

Cf. column k=1 A001651, A005773, A088855, column k=2 A247643, row sums A308435, A378810.

A088855(n,k) = {binomial(floor((n-1)/2), floor((k-1)/2))*binomial(ceil((n-1)/2),ceil((k-1)/2))}
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))))}
A_xy(10)

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)).

cp's OEIS Frontend

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

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