The odd part of (2^4)! is 3*(5*3*7)*(9*5*11*3*13*7*15) = 3^3*(5*7)^2*(9*11*13*15), which explains the Maple program below.
The table below shows, for e = 2..40, the 64 least significant bits of the odd part of (2^e)!, with the least significant bit at the left end, and with a space inserted immediately after the (e+1)st bit. For every row after the e=1 row, the first e+1 bits appear to have converged to their final values, and the (e+2)nd bit is the opposite of its apparent limiting value.
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e | 64 least significant bits of (2^e)! / 2^(2^e - 1)
---+------------------------------------------------------------------
2 | 110 0000000000000000000000000000000000000000000000000000000000000
3 | 1101 110010000000000000000000000000000000000000000000000000000000
4 | 11010 11101110111011100000110010000000000000000000000000000000000
5 | 110100 1011001110100011000001101010001001011100110001110101010100
6 | 1101000 000000001101000110100010110011110010011111011101100011000
7 | 11010001 10010101001010001010001101011100101011111100000011100010
8 | 110100010 0111100111001000110000010011101001010011011110111101110
9 | 1101000101 000110101110110000001011000011110010110111000000001101
10 | 11010001011 10101101100010111001010101001000111000001110000010111
11 | 110100010110 0010011111110101111010110111011111110111111000001001
12 | 1101000101101 110111110000001100111011011001001110011110011100011
13 | 11010001011010 11100001101100011101011000100010110100010111101000
14 | 110100010110100 1011001111011110110100001011000011010110110001101
15 | 1101000101101000 000011110100000100110000000000101000111010111100
16 | 11010001011010001 10000001011001001111011101000110001100111111001
17 | 110100010110100010 0101001000011101001001001110110111101010011110
18 | 1101000101101000101 011101001000000000010011011001011111000011010
19 | 11010001011010001011 00000101101001010101010110010100110101001110
20 | 110100010110100010111 1001101111101111101111001000000000100100001
21 | 1101000101101000101110 011011110000010111010010100110010110010001
22 | 11010001011010001011101 00100001101011100101101110111100001011110
23 | 110100010110100010111011 1101001111111011100011001000001111001111
24 | 1101000101101000101110110 111101110010100110111111001001110010010
25 | 11010001011010001011101100 10000001111101101100010011011100000001
26 | 110100010110100010111011000 0101001100111010011011010111011011000
27 | 1101000101101000101110110001 011101101100101111111110100110000000
28 | 11010001011010001011101100011 00000011011100010010011100001001001
29 | 110100010110100010111011000111 1001011000110111011010010000100011
30 | 1101000101101000101110110001110 011111001101100000001110000110110
31 | 11010001011010001011101100011101 00010101010011010011101000110011
32 | 110100010110100010111011000111011 1011101001111111010010100110011
33 | 1101000101101000101110110001110110 000111000010010001110010110011
34 | 11010001011010001011101100011101101 10100100111011011101001110011
35 | 110100010110100010111011000111011010 0011100100000001010100001011
36 | 1101000101101000101110110001110110101 111010101010011110010101011
37 | 11010001011010001011101100011101101010 10101101111010011001111011
38 | 110100010110100010111011000111011010100 0010011100001110100001111
39 | 1101000101101000101110110001110110101001 110111101011101010101000
40 | 11010001011010001011101100011101101010010 11100011110010101111010
<-------------- stable bits ------------->\<--- unstable bits ...
(End)
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