A379838 Triangle read by rows: T(n,k) is the total number of humps with height k in all Motzkin paths of order n, n >= 2 and 1 <= k <= n/2.

1, 3, 8, 1, 20, 5, 50, 19, 1, 126, 63, 7, 322, 196, 34, 1, 834, 588, 138, 9, 2187, 1728, 507, 53, 1, 5797, 5016, 1749, 253, 11, 15510, 14454, 5786, 1067, 76, 1, 41834, 41470, 18590, 4147, 416, 13, 113633, 118690, 58487, 15223, 1976, 103, 1, 310571, 339274, 181181, 53599, 8528, 635, 15

Offset: 2

Author

Xiaomei Chen, Jan 04 2025

Keywords

• nonn
• tabf

Examples

			Triangle begins:
   [2]     1;
   [3]     3;
   [4]     8,    1;
   [5]    20,    5;
   [6]    50,   19,   1;
   [7]   126,   63,   7;
   [8]   322,  196,  34,  1;
   [9]   834,  588, 138,  9;
  [10]  2187, 1728, 507, 53, 1;
  ...
		

Links

• Xiaomei Chen, Table of n, a(n) for n = 2..401
• Xiaomei Chen, Counting humps and peaks in Motzkin paths with height k, arXiv:2412.00668 [math.CO], Dec 2024.
• David He and Daniel Tubbenhauer, Tensor powers of representations of (diagram) monoids, arXiv:2508.04054 [math.RT], 2025. See p. 10.

Crossrefs

Row lengths give A004526.

Row sums give A097861.

Column 1 gives A140662.

Cf. A064189.

Programs

Sage

def A379838_triangel(dim):
    M = matrix(ZZ, dim, dim)
    for n in (2..dim+1):
        for k in (1..math.floor(n/2)+1):
            for i in range(n-2*k+1):
                if ((n-i)%2)==0:
                    M[n-2,k-1]=M[n-2, k-1]+(4*k)/(n-i+2*k)*binomial(n,i)*binomial(n-i-1,(n-i)/2+k-1)
    return M

Formula

G.f.: Sum_{n>=2, k>=1} T(n,k) * x^n * y^k = x^2 * M^2(x) * y / ((1-x) * (1 - x^2 * M^2(x) * y)), where M(x) is the g.f. for A001006.

T(n,k) = Sum_{i=0..n-2*k, i==n (mod 2)} (4*k) / (n-i+2*k) * binomial(n,i) * binomial(n-i-1,(n-i)/2+k-1).

T(n,k) = Sum_{i=2k-1..n-1} A064189(i,2k-1).

T(n,k) + T(n,k+1) = A064189(n,2k).