A379893 - OEIS

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

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OFFSET

0,8

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.

FORMULA

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).

T(n+1,2*k-1) + T(n,2*k-1) = A379838(n+1,k) - A379838(n,k).

EXAMPLE

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;

...

PROG

(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

CROSSREFS

Row sums give A257520.

Column 1 gives A005043.

Sequence in context: A368514 A289229 A263097 * A286011 A241954 A307047

Adjacent sequences: A379890 A379891 A379892 * A379894 A379895 A379896

KEYWORD

nonn,tabl

AUTHOR

Xiaomei Chen, Jan 05 2025

STATUS

approved