cp's OEIS Frontend

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.

%I A370396 #12 Mar 13 2024 12:13:49
%S A370396 1,3,13,63,347,2061,13219,89877,646009,4866339,38305573,313535631,
%T A370396 2661927255,23367856281,211680786375,1974332847177,18929186519705,
%U A370396 186249976522155,1878195826349765,19386702579997095,204603867473735387,2205553917952342605,24261717301000314867
%N 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.
%C A370396 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.
%e A370396 The a(2) = 13 matrices with sum of entries equal to 4:
%e A370396   [4]
%e A370396 .
%e A370396   [2 0] [1 1] [0 2]
%e A370396   [0 2] [1 1] [2 0]
%e A370396 .
%e A370396   [1 0 0] [0 0 1] [0 1 0]
%e A370396   [0 2 0] [0 2 0] [1 0 1]
%e A370396   [0 0 1] [1 0 0] [0 1 0]
%e A370396 .
%e A370396   [1 0 0 0] [0 0 0 1] [1 0 0 0]
%e A370396   [0 1 0 0] [0 1 0 0] [0 0 1 0]
%e A370396   [0 0 1 0] [0 0 1 0] [0 1 0 0]
%e A370396   [0 0 0 1] [1 0 0 0] [0 0 0 1]
%e A370396 .
%e A370396   [0 0 0 1] [0 1 0 0] [0 0 1 0]
%e A370396   [0 0 1 0] [1 0 0 0] [0 0 0 1]
%e A370396   [0 1 0 0] [0 0 0 1] [1 0 0 0]
%e A370396   [1 0 0 0] [0 0 1 0] [0 1 0 0]
%o A370396 (SageMath) nmax = 20
%o A370396 R.<x> = PowerSeriesRing(QQ)
%o A370396 S = [R(1)]
%o A370396 for k in range(nmax+1):
%o A370396     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])
%o A370396 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())
%Y A370396 Cf. A135401.
%K A370396 nonn
%O A370396 0,2
%A A370396 _Ludovic Schwob_, Feb 17 2024