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A368493 T(n,m) is the number of m-dimensional isotropic subspaces of a 2n-dimensional symplectic space over Z/2, n >= 0 and 0 <= m <= n.

%I A368493 #40 Dec 30 2023 03:08:50
%S A368493 1,1,3,1,15,15,1,63,315,135,1,255,5355,11475,2295,1,1023,86955,782595,
%T A368493 782595,75735,1,4095,1396395,50868675,213648435,103378275,4922775,1,
%U A368493 16383,22362795,3268162755,55558766835,112909751955,26883274275,635037975
%N A368493 T(n,m) is the number of m-dimensional isotropic subspaces of a 2n-dimensional symplectic space over Z/2, n >= 0 and 0 <= m <= n.
%C A368493 The number of m-dimensional isotropic subspaces of an n-dimensional symplectic space over Z/2.
%H A368493 J. Baez, <a href="https://math.ucr.edu/home/baez/week187.html">This Week's Finds in Mathematical Physics (Week 187)</a>
%H A368493 M. H. Poroch, <a href="https://doi.org/10.1142/S1793557119500694">Bounds on subspace codes based on totally isotropic subspace in symplectic spaces and extended symplectic spaces</a>, Asian-European Journal of Mathematics, 12 (2019).
%H A368493 Z. Wan, <a href="https://www.sciengine.com/Math%20A0/doi/10.1360/ya1964-13-3-515">Notes on finite geometries and the construction of PBIB designs I, Some Anzahl theorems in symplectic geometry over finite fields</a>, Acta Sci. 13 (1964) 515-516.
%F A368493 T(n,m) = Product_{i=n-m+1..n} (2^(2i)-1)/Product_{i=1..m} (2^i-1).
%e A368493 Triangle begins:
%e A368493   1;
%e A368493   1,    3;
%e A368493   1,   15,      15;
%e A368493   1,   63,     315,      135;
%e A368493   1,  255,    5355,    11475,      2295;
%e A368493   1, 1023,   86955,   782595,    782595,     75735;
%e A368493   1, 4095, 1396395, 50868675, 213648435, 103378275, 4922775;
%e A368493   ...
%t A368493 T[n_,m_]:=Product[(2^(2i)-1),{i,n-m+1,n}]/Product[(2^i-1),{i,1,m}]; Table[T[n,m],{n,0,7},{m,0,n}] (* _Stefano Spezia_, Dec 28 2023 *)
%o A368493 (Python)
%o A368493 from math import prod
%o A368493 q = 2
%o A368493 N = lambda n, m : (prod([q**(2*i)-1 for i in range(n-m+1, n+1)])//prod([q**i-1 for i in range(1, m+1)]))
%o A368493 print([N(n, m) for n in range(8) for m in range(n+1)])
%o A368493 (PARI) T(n,m) = prod(i=n-m+1, n, 2^(2*i)-1)/prod(i=1, m, 2^i-1); \\ _Michel Marcus_, Dec 27 2023
%Y A368493 Main diagonal gives A028362.
%Y A368493 Cf. A022166.
%K A368493 nonn,tabl
%O A368493 0,3
%A A368493 _Simon Burton_, Dec 27 2023