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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.

%I A379838 #59 Aug 11 2025 10:48:15
%S A379838 1,3,8,1,20,5,50,19,1,126,63,7,322,196,34,1,834,588,138,9,2187,1728,
%T A379838 507,53,1,5797,5016,1749,253,11,15510,14454,5786,1067,76,1,41834,
%U A379838 41470,18590,4147,416,13,113633,118690,58487,15223,1976,103,1,310571,339274,181181,53599,8528,635,15
%N 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.
%H A379838 Xiaomei Chen, <a href="/A379838/b379838.txt">Table of n, a(n) for n = 2..401</a>
%H A379838 Xiaomei Chen, <a href="https://arxiv.org/abs/2412.00668">Counting humps and peaks in Motzkin paths with height k</a>, arXiv:2412.00668 [math.CO], Dec 2024.
%H A379838 David He and Daniel Tubbenhauer, <a href="https://arxiv.org/abs/2508.04054">Tensor powers of representations of (diagram) monoids</a>, arXiv:2508.04054 [math.RT], 2025. See p. 10.
%F A379838 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.
%F A379838 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).
%F A379838 T(n,k) = Sum_{i=2k-1..n-1} A064189(i,2k-1).
%F A379838 T(n,k) + T(n,k+1) = A064189(n,2k).
%e A379838 Triangle begins:
%e A379838    [2]     1;
%e A379838    [3]     3;
%e A379838    [4]     8,    1;
%e A379838    [5]    20,    5;
%e A379838    [6]    50,   19,   1;
%e A379838    [7]   126,   63,   7;
%e A379838    [8]   322,  196,  34,  1;
%e A379838    [9]   834,  588, 138,  9;
%e A379838   [10]  2187, 1728, 507, 53, 1;
%e A379838   ...
%o A379838 (Sage)
%o A379838 def A379838_triangel(dim):
%o A379838     M = matrix(ZZ, dim, dim)
%o A379838     for n in (2..dim+1):
%o A379838         for k in (1..math.floor(n/2)+1):
%o A379838             for i in range(n-2*k+1):
%o A379838                 if ((n-i)%2)==0:
%o A379838                     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)
%o A379838     return M
%Y A379838 Row lengths give A004526.
%Y A379838 Row sums give A097861.
%Y A379838 Column 1 gives A140662.
%Y A379838 Cf. A064189.
%K A379838 nonn,tabf
%O A379838 2,2
%A A379838 _Xiaomei Chen_, Jan 04 2025