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.
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, 34, 15, 6, 0, 1, 91, 141, 139, 99, 55, 21, 7, 0, 1, 232, 371, 379, 293, 175, 83, 28, 8, 0, 1, 603, 982, 1043, 847, 551, 286, 119, 36, 9, 0, 1, 1585, 2628, 2872, 2441, 1684, 956, 441, 164, 45, 10, 0, 1
Offset: 0
Examples
Triangle begins: [0] 1; [1] 0, 1; [2] 1, 0, 1; [3] 1, 2, 0, 1; [4] 3, 3, 3, 0, 1; [5] 6, 9, 6, 4, 0, 1; [6] 15, 21, 19, 10, 5, 0, 1; [7] 36, 55, 50, 34, 15, 6, 0, 1; [8] 91, 141, 139, 99, 55, 21, 7, 0, 1; ...
Links
- Xiaomei Chen, Table of n, a(n) for n = 0..860
- Xiaomei Chen, Counting humps and peaks in Motzkin paths with height k, arXiv:2412.00668 [math.CO], Dec 2024.
Programs
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Sage
def A379893_triangel(dim): M = matrix(ZZ, dim, dim) for n in range(dim): for k in range(n+1): for i in range(math.floor((n-k-1)/2)+1): for j in range(n-k-1-2*i+1): if ((n+k-1-j)%2)==0: 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) M[n,k]=M[n,k]-pow(-1,n+k+1) return M