Harmonic Number Sum Congruence

[Robert G. Wilson v]

Dear Leroy,

	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}]


Sincerely,

Bob.


Leroy Quet wrote:
> I forgot to post the following to seq.fan:
> 
> By the way, the sequence of 
> 
> {m! h(m) (mod {m+1})}  -> 1, 1 ,0, 3, 4, 2,.. (?)
> 
> does not seem to be in the EIS either.
> 
> Is there a closed form for this sequence? ...
> 
> (...which also equals
> {sum{k=1 to m}  H(k) k! (m-k)! (mod {m+1})})
> 
> 
> Regarding:
> 
>>I wrote:
>>
>>
>>>Below is some of a post I made recently to sci.math.
>>>
>>>I have added some more seq.fan-friendly material and edited the rest of 
>>>my original (truncated) message.
>>>
>>>
>>>
>>>
>>>Let H(n) = sum{k=1 to n} 1/k, 
>>>the n_th harmonic number.
>>>
>>>I suppose that, for each m = positive integer,
>>>
>>>
>>>
>>>sum{k=1 to m-1}  H(k) k! (m-k)!
>>>
>>>is congruent to
>>>
>>>(1/2) m! H(floor(m/2))) (-1)^m 
>>>
>>>(mod (m+1)) .
>>>
>>>
>>>(I do not know if the result is trivial {or am certain it is correct}.)
>>>
>>
>>
>>
>>
>>We can rewrite the above so it becomes, perhaps to some reading this, 
>>more simply stated and more natural.
>>
>>(Take sum from 1 to m, instead of from 1 to m-1)
>>
>>
>>sum{k=1 to m}  H(k) k! (m-k)!
>>
>>
>>
>>is congruent to
>>
>>
>>
>>m! *h(m)   (mod(m+1)),
>>
>>
>>where h(m) = sum{k=0 to floor((m-1)/2)}  1/(m -2k)  .
>>
>>
>>
>>Now, h(m) =
>>sum{k=0 to floor((m-1)/2)}  1/(m -2k)
>>is interesting in itself, for it is (analogously to "double-factorials") 
>>the sum of the reciprocal of EVERY-OTHER positive integer <=m 
>>(even or odd, depending on m).
>>
>>
>>Neither sequence:
>>
>>m!*h(m) 
>>nor
>>m!!*h(m)
>>
>>nor
>>sum{k=1 to m}  H(k) k! (m-k)!
>>
>>seems to be in the EIS either, if I calculated correctly by hand.
>>
>>Thanks,
>>Leroy
>>Quet
>>
>>
>>>I decided to post this message to seq.fan because the sequence formed from
>>>sum{k=1 to m-1}  H(k) k! (m-k)!
>>>is not in the database.
>>>
>>>It would be more natural too take the sum to m instead of (m-1).
>>>So, the sequence
>>>sum{k=1 to m}  H(k) k! (m-k)!
>>>begins (figured by hand):
>>>
>>>1, 4, 16, 73, 388, ...
>>>
>>>And, no, this sequence does not seem to be in the EIS either.
>>>
>>>
>>>
>>>thanks,
>>>Leroy Quet
> 
>