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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. 0
1, 1, 3, 1, 15, 15, 1, 63, 315, 135, 1, 255, 5355, 11475, 2295, 1, 1023, 86955, 782595, 782595, 75735, 1, 4095, 1396395, 50868675, 213648435, 103378275, 4922775, 1, 16383, 22362795, 3268162755, 55558766835, 112909751955, 26883274275, 635037975 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The number of m-dimensional isotropic subspaces of an n-dimensional symplectic space over Z/2.
LINKS
FORMULA
T(n,m) = Product_{i=n-m+1..n} (2^(2i)-1)/Product_{i=1..m} (2^i-1).
EXAMPLE
Triangle begins:
1;
1, 3;
1, 15, 15;
1, 63, 315, 135;
1, 255, 5355, 11475, 2295;
1, 1023, 86955, 782595, 782595, 75735;
1, 4095, 1396395, 50868675, 213648435, 103378275, 4922775;
...
MATHEMATICA
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 *)
PROG
(Python)
from math import prod
q = 2
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)]))
print([N(n, m) for n in range(8) for m in range(n+1)])
(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
CROSSREFS
Main diagonal gives A028362.
Cf. A022166.
Sequence in context: A178657 A257490 A156289 * A095922 A263632 A284861
KEYWORD
nonn,tabl
AUTHOR
Simon Burton, Dec 27 2023
STATUS
approved

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Last modified May 18 12:18 EDT 2024. Contains 372630 sequences. (Running on oeis4.)