[seqfan] Re: doubtful relevance
Alonso Del Arte
alonso.delarte at gmail.com
Sun Jan 24 01:09:37 CET 2010
Well, Wouter, for what it's worth, I think you should submit that sequence,
but perhaps put it in a list of sequences to submit once the transition to
the wiki is complete. The powers of 3, whether in sequence or scrambled,
turn up surprisingly in a few different contexts, even when the explanation
turns out to seem rather mundane. I would bet money that lots of people, in
the course of investigating various problems, will come up with various
sequences of powers of 3.
Al
On Sat, Jan 23, 2010 at 9:59 AM, wouter meeussen <wouter.meeussen at pandora.be
> wrote:
> dear All,
>
> for the following sequence, I'm really in two minds whether to submit it or
> not.
> I dislike submitting sequences that will never get hit on.
> Should I do it anyhow?
> Just imagine the ugly sequence description that would make!
> -----
> Firstly, I put myself the problem of calculating <Y(1,0)^n , Y(n,0)>^2
> efficiently.
> Here the bra-ket notation < .. , .. > symbolises
> Integrate[ SphericalHarmonicY[1,0,th,fi]^n *
> SphericalHarmonicY[n,0,th,-1*fi] Sin[th],{th,0,Pi},{fi,0,2Pi}]
> which is, I admit, a rather arbitrary but somewhat esthetic expression.
> Apart from a factor Pi^(n-1), this comes out as
> a(n=1,2,...) = 1, 1/5, 27/700, 9/1225, 3/2156, 81/308308, ...
>
> What's cute is that the numerators are exact powers of 3:
> 3^{0, 0, 3, 2, 1, 4, 4, 4, 9, 9, 9, 12, 10, 8, 11, 11, 11, 16, 16, 16, 19,
> 18, 17,
> 20, 20, 20, 27, 27, 27, 30, 29, 28, 31, 31, 31, 36, 36, 36, 39, 36, 33, 36,
> 36, 36, 41, 41, 41, 44, 43, 42, 45, 45, 45, 52, 52, 52, 55, 54, 53, 56, 56,
> 56, 61, 61,...}
> for which Superseeker comes up blank.
> The denominators consist of products of small primes, less than 2n.
> It contains powers of 2 in the form 2^(-2+ 2 * A000120(n))
>
> But there is no mystery involved, just repeated evaluation of the
> well-known
> Sqrt[(2a+1)(2b+1)(2a+2b+1)/4/Pi]*ThreeJSymbol[{a,0},{b,0},{a+b,0}]^2
> say w[a,b], at specific values of a and b:
>
> {1/5, 1/5*w[1, 2]^2, 1/25*w[2, 2]^2, 1/25*w[1, 4]^2*w[2, 2]^2, 1/125*w[2,
> 2]^2*w[2, 4]^2,
> 1/125*w[1, 6]^2*w[2, 2]^2*w[2, 4]^2, 1/625*w[2, 2]^4*w[4, 4]^2, 1/625*w[1,
> 8]^2*w[2, 2]^4*
> w[4, 4]^2, (w[2, 2]^4*w[2, 8]^2*w[4, 4]^2)/3125, (w[1, 10]^2*w[2,
> 2]^4*w[2, 8]^2*w[4, 4]^2)/3125,
> (w[2, 2]^6*w[4, 4]^2*w[4, 8]^2)/15625, (w[1, 12]^2*w[2, 2]^6*w[4,
> 4]^2*w[4, 8]^2)/15625}
>
> how artificial is <Y(1,0)^n , Y(n,0)>^2 Pi^(n-1) to your taste?
>
> Wouter.
>
>
>
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