Different patterns in Sn modulo 2, is it A001405?

[franktaw at netscape.net]

I'm assuming, based on your example, that by "different binary
patterns", you mean different patterns modulo 2.

This is indeed the same as A001405, and it is easy to see why.

Of the numbers from 1 to n, half (rounded up) are odd, and the
other half (rounded down) are even.  So there will be ceiling(n/2)
1's, and floor(n/2) 0's in the final pattern.  Now you just have to
decide which places will be 0's.  Floor(n/2) objects can be
arranged in n positions in exactly C(n,floor(n/2)) ways -- this is
what the "C" (combinations) function means.  And, obviously,
any such arrangement can be achieved using an appropriate
permutation, since we can put the numbers anywhere.

There is no question that the definition in A001405 is
mathematically simpler than your formulation using S_n.
The combinations function is one of the most fundamental in
combinatorics, and while S_n is equally fundamental, looking
at numbers modulo 2 is a fairly specialized operation.

Franklin T. Adams-Watters

-----Original Message-----
From: Ivica Kolar <telpro at kvid.hr>

Sequence definition:
Different binary patterns
in the set of all permutations Sn taken modulo 2.
Example: 0,1,2,3,4 modulo 2 = 0,1,0,1,0
Short calculation gives
sequence:
;N
0  1  2  3  4   5    6    7   8    9
;   1, 2, 3, 6, 10, 20, 35, 70, 126
;A001405: 1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, 924, 1716

...

Sequence definition over Sn seems simplest than definition(s) given in 
OEIS,
from my point of view at least.
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