[VA] SRC#001 - Spiky Integral

[Gerson W. Barbosa]

(07-19-2018 12:27 AM)Thomas Klemm Wrote:  

(07-18-2018 05:32 PM)ijabbott Wrote:  Is there a neat formula for just the constant term when converting the product to a sum?

From A058377:

Quote:FORMULA
a(n) is half the coefficient of q^0 in product('(q^(-k)+q^k)', 'k'=1..n) for n >= 1.

Thus I doubt there is a "neat formula".

There is an asymptotic formula, but it doesn't help much:

https://cs.uwaterloo.ca/journals/JIS/VOL...ivan8.html

(((n^2+n)/2+1) mod 2)*sqrt(6/pi)*2^(n-1)/(n*sqrt(n))

n = 3 -> 1.06385 (1)
n = 4 -> 1.38198 (1)
n = 8 -> 7.81764 (7)
n = 11 -> 38.7893 (35)
n = 27 -> 661051 (632602)
n = 1000 -> 2.34135e296 (2.3385429e296)


A small correction might help a bit:

(((n^2+n)/2+1) mod 2)* sqrt(6/pi)*2^(n-1)*(1-6/(5*n)+21/(20*n^2)-1/(8*n^3)+3/n^4)/(n*sqrt(n))

n = 3 ->0.7969 (1)
n = 4 -> 1.07157 (1)
n = 8 -> 6.77707 (7)
n = 11 -> 34.8986 (35)
n = 27 -> 632623 (632602)
n = 1000 -> n = 1000 -> 2.33854293231e296 (2.33854293496e296)

Regards,

Gerson.