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

2,2

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.

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).

Example

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;
  ...

Prog

(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

Crossrefs

Row lengths give A004526.

Row sums give A097861.

Column 1 gives A140662.

Cf. A064189.

Sequence in context: A197725 A288875 A152230 * A181371 A118357 A278866

Adjacent sequences: A379834 A379835 A379837 * A379839 A379840 A379841

Keyword

nonn,tabf

Author

Xiaomei Chen, Jan 04 2025

Status

approved