# [seqfan] doubtful relevance

wouter meeussen wouter.meeussen at pandora.be
Sat Jan 23 15:59:20 CET 2010

```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.

```