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A379893 Triangle read by rows: T(n,k) is the number of standard Young tableaux with shapes in {lambda = (lambda_1,lambda_2,...) | lambda_1-lambda_2=k, lambda_i<=1 for i>=3, |lambda| = n}, n >= 0 and 0 <= k <= n.

%I A379893 #34 Jan 15 2025 09:02:44
%S A379893 1,0,1,1,0,1,1,2,0,1,3,3,3,0,1,6,9,6,4,0,1,15,21,19,10,5,0,1,36,55,50,
%T A379893 34,15,6,0,1,91,141,139,99,55,21,7,0,1,232,371,379,293,175,83,28,8,0,
%U A379893 1,603,982,1043,847,551,286,119,36,9,0,1,1585,2628,2872,2441,1684,956,441,164,45,10,0,1
%N A379893 Triangle read by rows: T(n,k) is the number of standard Young tableaux with shapes in {lambda = (lambda_1,lambda_2,...) | lambda_1-lambda_2=k, lambda_i<=1 for i>=3, |lambda| = n}, n >= 0 and 0 <= k <= n.
%H A379893 Xiaomei Chen, <a href="/A379893/b379893.txt">Table of n, a(n) for n = 0..860</a>
%H A379893 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.
%F A379893 T(n,k) = (-1)^(n+k) + Sum_{i=0..(n-k-1)/2} Sum_{j=0..n-k-1-2*i, j==n+k-1 (mod 2)} (2*k+2) / (n+k+1-2*i-j) * binomial(n-2*i-2,j) * binomial(n-2*i-j-1,(n+k-j-1)/2-i).
%F A379893 T(n+1,2*k-1) + T(n,2*k-1) = A379838(n+1,k) - A379838(n,k).
%e A379893 Triangle begins:
%e A379893   [0]   1;
%e A379893   [1]   0,   1;
%e A379893   [2]   1,   0,   1;
%e A379893   [3]   1,   2,   0,  1;
%e A379893   [4]   3,   3,   3,  0,  1;
%e A379893   [5]   6,   9,   6,  4,  0,  1;
%e A379893   [6]  15,  21,  19, 10,  5,  0, 1;
%e A379893   [7]  36,  55,  50, 34, 15,  6, 0, 1;
%e A379893   [8]  91, 141, 139, 99, 55, 21, 7, 0, 1;
%e A379893   ...
%o A379893 (Sage)
%o A379893 def A379893_triangel(dim):
%o A379893     M = matrix(ZZ, dim, dim)
%o A379893     for n in range(dim):
%o A379893         for k in range(n+1):
%o A379893             for i in range(math.floor((n-k-1)/2)+1):
%o A379893                 for j in range(n-k-1-2*i+1):
%o A379893                     if ((n+k-1-j)%2)==0:
%o A379893                         M[n,k]=M[n, k]+(2*k+2)/(n+k+1-2*i-j)*binomial(n-2*i-2,j)*binomial(n-2*i-j-1,(n+k-j-1)/2-i)
%o A379893             M[n,k]=M[n,k]-pow(-1,n+k+1)
%o A379893     return M
%Y A379893 Row sums give A257520.
%Y A379893 Column 1 gives A005043.
%K A379893 nonn,tabl
%O A379893 0,8
%A A379893 _Xiaomei Chen_, Jan 05 2025