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[Richard Mathar]

lq> From seqfan-owner at ext.jussieu.fr  Sun Oct 22 18:39:02 2006
lq> Return-Path: <seqfan-owner at ext.jussieu.fr>
lq> Subject: Re: Ugly But Interesting Harmonic Number Sequence
lq> Date: Sun, 22 Oct 06 10:33:44 -0600
lq> From: Leroy Quet <qq-quet at mindspring.com>
lq> cc: <seqfan at ext.jussieu.fr>
lq> ...
lq> The result is easily derived from the earlier result of mine:
lq> (View with fixed-width font.)
lq> 
lq>  n
lq> ---
lq> \   (H(2k) - H(n-k)) (2k)!
lq> /   ----------------------- = 0,
lq> ---      k! (k+n+1)!
lq> k=0
lq> 
lq> for all nonnegative integers n.
lq> 
lq> (H(m) = sum{j=1 to m} 1/j, the mth harmonic number. H(0) = 0.)
lq> 
lq> It has been some time since I found the above result. And, since I cannot 
lq> at this time reverse-engineer a proof, I am not ABSOLUTELY certain that 
lq> it is correct.
lq> 
lq> In any case, I just submitted the following:
lq> 
lq> >%I A124235
lq> >%S A124235 1,1,17,877
lq> >%N A124235 a(n) = numerator of (sum{k=1 to n} H(2k)(2k)!/(k!(k+n+1)!) = 
lq> >sum{k=0 to n-1} H(n-k)(2k)!/(k!(k+n+1)!)), where H(k) = sum{j=1 to k} 1/j 
lq> >(ie the kth harmonic number).
lq> >%Y A124235 A124236
lq> >%O A124235 0
lq> >%K A124235 ,frac,more,nonn,
lq> >%A A124235 Leroy Quet (qq-quet at mindspring.com), Oct 22 2006
lq> 
lq> >%I A124236
lq> >%S A124236 2,3,144,30240
lq> >%N A124236 a(n) = denominator of (sum{k=1 to n} H(2k)(2k)!/(k!(k+n+1)!) = 
lq> >sum{k=0 to n-1} H(n-k)(2k)!/(k!(k+n+1)!)), where H(k) = sum{j=1 to k} 1/j 
lq> >(ie the kth harmonic number).
lq> >%Y A124236 A124235
lq> >%O A124236 1
lq> >%K A124236 ,frac,more,nonn,
lq> >%A A124236 Leroy Quet (qq-quet at mindspring.com), Oct 22 2006
lq> 
lq> My request is to whomever extends the sequences: Once the sequences 
lq> appear in the database, could you please try the values for both sums, 
lq> just as a check that the sums are the same for the first N values of n at 
lq> least?

I tested this for up to n=30 and the two expressions are the same
numerically (as shown by the two lists below).
lq> 
lq> thanks,
lq> Leroy Quet
lq> 
lq> PS: There are a few seq.fan people who could EASILY prove that the two 
lq> sums equal each other, I bet.
lq> 

The PARI program is (with the two "alt" versions evaluating the alternative
expressions):

H(n)={
	if(n==0,
		0,
		sum(k=1,n,1/k)
	) ;
}
A124235(n)={
	numerator(sum(k=1,n,H(2*k)*(2*k)!/k!/(k+n+1)!)) ;
}
A124235alt(n)={
	numerator(sum(k=0,n-1,H(n-k)*(2*k)!/k!/(k+n+1)!)) ;
}
A124236(n)={
	denominator(sum(k=1,n,H(2*k)*(2*k)!/k!/(k+n+1)!)) ;
}
A124236alt(n)={
	denominator(sum(k=0,n-1,H(n-k)*(2*k)!/k!/(k+n+1)!)) ;
}
{
	for(n=1,30,
		print(n," ",A124235(n)," ",A124236(n)) ;
	) ;
	for(n=1,30,
		print(n," ",A124235alt(n)," ",A124236alt(n)) ;
	) ;
}

and the results for n, A124235 and A124236 side by side for n=1..30 look as follows
(for anyone to submit them as extensions):

1 1 2
2 1 3
3 17 144
4 877 30240
5 26 4725
6 6827 7983360
7 12310607 108972864000
8 105059 8072064000
9 604489 453682944000
10 49568347 403179783552000
11 12933671 1250891123328000
12 143562866581 179527894020034560000
13 2406858923083 42009527200688087040000
14 35714915113 9335450489041797120000
15 530084035699 2215047797854462771200000
16 7390807289267 524756481458026727669760000
17 1031992153425439 1318304893973998256957030400000
18 225749374968517 5476035713430454298129203200000
19 8052704479475951909 3903548250033191902787017467494400000
20 127959798438059449 1301182750011063967595672489164800000
21 716400163049690921 160045478251360868014267716167270400000
22 603716771399416157789 3096880004163832796076080307836682240000000
23 193227662249524793923 23742746698589384769916615693414563840000000
24 111558623417695906519 341947520167919347274530957620066385920000000
25 1603445713588619416794797 127481454993802011857417886310336949334835200000000
26 42003215512683127303 89934007050301242932922671118403491594240000000
27 11776478817476869934097997 704100204581557006543022788916145234905137152000000000
28 6174583834198133825389588823 10675567301865567333205311525546594051631689498624000000000
29 128738271275714955211769263 6657880682883902207805463101953789838652021407744000000000
30 6530130657597274436349122893 10437654659138277546922450297262984241338183275511808000000000
1 1 2
2 1 3
3 17 144
4 877 30240
5 26 4725
6 6827 7983360
7 12310607 108972864000
8 105059 8072064000
9 604489 453682944000
10 49568347 403179783552000
11 12933671 1250891123328000
12 143562866581 179527894020034560000
13 2406858923083 42009527200688087040000
14 35714915113 9335450489041797120000
15 530084035699 2215047797854462771200000
16 7390807289267 524756481458026727669760000
17 1031992153425439 1318304893973998256957030400000
18 225749374968517 5476035713430454298129203200000
19 8052704479475951909 3903548250033191902787017467494400000
20 127959798438059449 1301182750011063967595672489164800000
21 716400163049690921 160045478251360868014267716167270400000
22 603716771399416157789 3096880004163832796076080307836682240000000
23 193227662249524793923 23742746698589384769916615693414563840000000
24 111558623417695906519 341947520167919347274530957620066385920000000
25 1603445713588619416794797 127481454993802011857417886310336949334835200000000
26 42003215512683127303 89934007050301242932922671118403491594240000000
27 11776478817476869934097997 704100204581557006543022788916145234905137152000000000
28 6174583834198133825389588823 10675567301865567333205311525546594051631689498624000000000
29 128738271275714955211769263 6657880682883902207805463101953789838652021407744000000000
30 6530130657597274436349122893 10437654659138277546922450297262984241338183275511808000000000

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