[seqfan] Re: Definition of A002190

[Charles Greathouse]

Agreed.  The formula in A101981 should also change to ...for n >= 0.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Sat, Oct 9, 2010 at 7:03 PM, Alois Heinz <heinz at hs-heilbronn.de> wrote:
>
> The offset should be 0 with a(0)=0, a(1)=1, a(2)=1, ...
>
> Maple code:
>
> a:= n-> coeff (series (-ln(BesselJ(0,2*sqrt(x))), x, n+1), x, n)*(n!)^2;
> seq (a(n), n=0..30);
>
> gives:
>
> 0,1,1,4,33,456,9460,274800,10643745,530052880,32995478376, ...
>
> see also the signed version A101981
>
> Alois
>
> Am 09.10.2010 23:52, schrieb Paul D Hanna:
>> Seqfans,
>>        The definition of A002190 is:
>> Sum_{n>=0} a(n)*x^n/n!^2 = -ln(BesselJ(0,2*sqrt(x))).
>> [1, 4, 33, 456, 9460, 274800, 10643745, 530052880, 32995478376, 2510382661920, ...]
>> (OFFSET 2).
>>
>> However, by the definition the offset should be 1 and a(1) = 1, since
>> -log( Sum_{n>=0} (-x)^n/n!^2 ) = x + x^2/2!^2 + 4*x^3/3!^2 + 33*x^4/4!^2 + 456*x^5/5!^2 + 9460*x^6/6!^2 +...
>>
>> The question is, should the sequence be changed to include a(1)=1,
>> or should the definition be changed to fit the existing sequence, i.e.,
>> change the name to:
>>     Sum_{n>=0} a(n)*x^n/n!^2 = -x-ln(BesselJ(0,2*sqrt(x))).
>>
>> I would vote for prepending a '1' to the existing sequence and changing OFFSET=1 to fit the definition,
>> but I fear changing an old classic sequence without the consensus of SeqFans.
>>
>> Thanks,
>>        Paul
>
>
>
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