|
|
A053722
|
|
Number of n X n binary matrices of order dividing 2 (also number of solutions to X^2=I in GL(n,2)).
|
|
25
|
|
|
1, 4, 22, 316, 6976, 373024, 32252032, 6619979776, 2253838544896, 1810098020122624, 2442718932612677632, 7758088894129169760256, 41674675294431186817908736, 526370120583359572695165435904
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
In characteristic 2, A^2 = I if and only if B^2 = 0 where B = I + A, so a(n) is also equal to the number of n X n binary matrices B such that B^2 = 0.
Conjecture: the two matrices I and 0 have the largest number of square roots. Checked for n=1..5. - Alexey Slizkov, Jan 11 2024
|
|
REFERENCES
|
Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{k=1..floor(n/2)} (2^n - 1)(2^n - 2) ... (2^n - 2^{n-k+1})/(2^k - 1)(2^k - 2)....(2^k - 2^{k-1}) * (2^{n-k} - 1)(2^{n-k} - 2)...(2^{n-k} - 2^{n-2k+1}). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 05 2001
|
|
MAPLE
|
Q:= Product(1+u/2^i, i=1..infinity)/Product(1-u^2/2^i, i=1..infinity):
S:= series(Q, u, 31):
seq(coeff(S, u, n)*mul(2^i-1, i=1..n), n=1..30); # Robert Israel, Mar 26 2018
|
|
MATHEMATICA
|
QP = QPochhammer; Q = (1-x) QP[-x, 1/2]/QP[x^2, 1/2];
Table[(-1)^n QP[2, 2, n] SeriesCoefficient[Q, {x, 0, n}], {n, 1, 14}] (* Jean-François Alcover, Sep 17 2018, from Maple *)
|
|
PROG
|
# (Sage)
g = lambda n: GL(n, 2).order() if n>0 else 1
a053722 = lambda n: g(n)*sum(1/(g(k)*g(n-2*k)*2**(k**2+2*k*(n-2*k))) for k in range(1+floor(n/2))) if n>0 else 0
map(a053722, range(25))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|