Harmonic Number Sum Congruence

[Leroy Quet]

Robert G. Wilson wrote:
>	I get:
>1, 4, 16, 73, 388, 2396, 17024, 137544, 1248816, 12603288, 140018688, 
>1698063552, 
>22318009344, 315942698880, 4791898275840, 77510315197440, 1331759355586560, 
>24220225133061120, 464796175236710400, 9385769913543475200, ..., using:
>Table[ Sum[ HarmonicNumber[k]k!(m - k)!, {k, 1, m}], {m, 1, 20}]
>
>	Whereas the original seq:
>1, 1, 0, 3, 4, 2, 0, 6, 6, 5, 0, 3, 8, 0, 0, 13, 0, 3, 0, 0, 12, 17, 0, 0, 
>14, 0, 
>0, 1, 0, 6, 0, 0, 18, 0, 0, 1, 20, 0, 0, 23, 0, 25, 0, 0, 24, 44, 0, 0, 0, 
>0, 0, 
>36, 0, 0, 0, 0, 30, 8, 0, 36, 32, 0, 0, 0, 0, 10, 0, 0, 0, 2, ..., using:
>Table[ Mod[ Sum[ HarmonicNumber[k]k!(m - k)!, {k, 1, m}], m + 1], {m, 1, 70}]

Thanks for figuring above.

I realize now that the zeros in the mod-sequence become more frequent.

For instance, if (m+1)|(m-1)!!  
((m-1)!!    = (m-1)(m-3)(m-5)...),
then
the corresponding term in the mod-sequence is zero.
(And it MAY be zero in additional situations too.)

thanks,
Leroy Quet