A118178 integral in closed form

[Max]

On 4/18/06, Max <maxale at gmail.com> wrote:

> I have not got the closed form yet but I found the following
> connection to the sequence A092765. The integral in question is equal
> to
>
> 2 * Pi * SUM[n=0..+oo] binomial(1/2,n) * 4^(-n) * A092765(n)
>
> I am going to submit some clarifying comments to A092765.
>
> btw, there is a missing integral sign in the formula (11) at
> http://mathworld.wolfram.com/EightCurve.html

I have just submitted the following comments to A092765:

%S A092765 1, 0, 4, 6, 36, 100, 430, 1470, 5796, 21336, 82404, 312180,
1203246, 4617756, 17846686, 68974906, 267498660, 1038555024,
4040525320, 15739195680, 61399048036, 239788778760, 937536139764,
3669179504364, 14373144873774, 56350223472600, 221094286028100,
868099633603800, 3410759865958110, 13409152861537860,
52747600247673930
%F A092765 a(n) = 2^(2n+1) / Pi * Int(cos(t)^n*cos(3*t)^n, t=0..Pi/2)
a(n) = Sum(binomial(n,k)*binomial(4*n-2*k,2*n-k)*(-3)^k, k=0..n)
G.F.: (1 + sqrt(1-4*x)) / ( sqrt(1-4*x) * (
sqrt(1+6*x+2*sqrt(9*x2+4*x)) + sqrt(1+6*x-2*sqrt(9*x2+4*x)) ) )
%o A092765 (PARI) a(n) = sum(k=0,n,binomial(n,k)*binomial(4*n-2*k,2*n-k)*(-3)^k)

The formula
a(n) = 2^(2n+1) / Pi * Int(cos(t)^n*cos(3*t)^n, t=0..Pi/2)
together with the formula (11) at
http://mathworld.wolfram.com/EightCurve.html give rise to the formula
A118178  = 2 * Pi * SUM[n=0..+oo] binomial(1/2,n) * 4^(-n) * A092765(n)
that I mentioned in the previous message.

Max