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A370396
Number of nonnegative integer matrices with sum of entries equal to 2*n or 2*n+1, no zero rows or columns, which are symmetric about both diagonals.
1
1, 3, 13, 63, 347, 2061, 13219, 89877, 646009, 4866339, 38305573, 313535631, 2661927255, 23367856281, 211680786375, 1974332847177, 18929186519705, 186249976522155, 1878195826349765, 19386702579997095, 204603867473735387, 2205553917952342605, 24261717301000314867
OFFSET
0,2
COMMENTS
a(n) is the number of semistandard Young tableaux of size 2*n or 2*n+1 with consecutive entries (i.e., if i is in T, and 1<=j<=i, then j is in T) which are invariant under Schützenberger involution.
EXAMPLE
The a(2) = 13 matrices with sum of entries equal to 4:
[4]
.
[2 0] [1 1] [0 2]
[0 2] [1 1] [2 0]
.
[1 0 0] [0 0 1] [0 1 0]
[0 2 0] [0 2 0] [1 0 1]
[0 0 1] [1 0 0] [0 1 0]
.
[1 0 0 0] [0 0 0 1] [1 0 0 0]
[0 1 0 0] [0 1 0 0] [0 0 1 0]
[0 0 1 0] [0 0 1 0] [0 1 0 0]
[0 0 0 1] [1 0 0 0] [0 0 0 1]
.
[0 0 0 1] [0 1 0 0] [0 0 1 0]
[0 0 1 0] [1 0 0 0] [0 0 0 1]
[0 1 0 0] [0 0 0 1] [1 0 0 0]
[1 0 0 0] [0 0 1 0] [0 1 0 0]
PROG
(SageMath) nmax = 20
R.<x> = PowerSeriesRing(QQ)
S = [R(1)]
for k in range(nmax+1):
S.append(sum(S[i]*binomial(k, i)*x^(2*(k-i)) for i in range(k+1))/(1-x^2+O(x^(nmax+1)))^k/(1-x+O(x^(nmax+1)))-S[k])
print(sum(1/(1-x+O(x^(nmax+1)))/(1-x^2+O(x^(nmax+1)))^n*sum(x^(2*(n-k))*factorial(n)/factorial(n-k)/factorial(k-i)/factorial(k-j)/factorial(i+j-k)*S[i]*S[j] for k in range(n+1) for i in range(k+1) for j in range(k-i, k+1)) for n in range(nmax+1)).coefficients())
CROSSREFS
Cf. A135401.
Sequence in context: A284716 A107097 A202837 * A180111 A292183 A006923
KEYWORD
nonn
AUTHOR
Ludovic Schwob, Feb 17 2024
STATUS
approved