Ugly But Interesting Harmonic Number Sequence

[Richard Mathar]

lq> From seqfan-owner at ext.jussieu.fr  Sat Oct 21 21:39:15 2006
lq> Subject: Ugly But Interesting Harmonic Number Sequence
lq> Date: Sat, 21 Oct 06 13:33:52 -0600
lq> From: Leroy Quet <qq-quet at mindspring.com>
lq> To: <seqfan at ext.jussieu.fr>
lq> 
lq> Let H(n) = sum{k=1 to n} 1/k, the nth harmonic number.
lq> (H(0) = 0.)
lq> 
lq> Then, for n = any *positive* integer,
lq> 
lq> a(n) = ( (2n)!*(1 -H(2n))/2 + (2n+1)!*(H(n+1)/(n+2) - H(2n+2)/(2n+3)) 
lq> )/(n+1)
lq> 
lq> is always an integer.
lq> 
lq> 
lq> I based this result on an identity I found years ago which I cannot 
lq> remember how I got it. So perhaps the above result is wrong.
lq> 
lq> But my question to seq.fan is, is it appropriate to submit unaestheticly 
lq> generated integer sequences which are interesting solely because every 
lq> term is an integer?
lq> 
lq> And if so, and if each a(n) is an integer, could someone please calculate 
lq> and submit the sequence {a(k)}?
lq> 
lq> Thanks,
lq> Leroy Quet

The first 20 terms (n=1,2,...20) are most likely

0
0
14
1368
157080
23523552
4587589440
1143034260480
356276193223680
136251924996096000
62851092138616320000
34455124535769907200000
22160020972109948928000000
16534267800892097159135232000
14171787174978077671563264000000
13833174876913304034850520432640000
15259347383833047535006261130035200000
18892410244840363651282496875659264000000
26091816893840033826803262856392553267200000
39974120611844772536714822349955792896000000000

The PARI program for these is [there is no need to define H(0)...]
H(n)={
        return( sum(k=1,n,1/k) ) ;
}
a(n)={
        return( ((2*n)!*(1-H(2*n))/2+(2*n+1)!*(H(n+1)/(n+2)-H(2*n+2)/(2*n+3)))/(n+1) ) ;
}

{
        for(n=1,20,
                print(a(n)) ;
        ) ;
}

Of course I would never submit ugly integer sequences, unless there
would be some hidden beauty in there.. -:)

RJ Mathar, http://www.strw.leidenuniv.nl/~mathar