A138896 Ratio of (2n-1)! to number of zeros in Sylvester matrix of polynomial of n degree with all nonzero coefficients.

3, 15, 280, 11340, 798336, 86486400, 13343616000, 2778808032000, 750895681536000, 255454710858547200, 106826515449937920000, 53858368206010368000000, 32215590089995124736000000, 22555515290152300904448000000, 18272974787062050706056806400000, 16959604724241965811558973440000000

Offset

2,1

Comments

(2n-1)! = A009445(n-1) is the number of monomials in determinant of symbolic square matrix of size 2n-1 X 2n-1 without zeros.

Denominators in the series expansion of (1/2)*(Pi/(2*x))^(1/2)* (x*BesselI(1/2, x) - BesselI(3/2, x)). - Abdallah Daddi-Moussa-Ider, Jul 25 2024

Links

Table of n, a(n) for n=2..17.

Formula

a(n) = (2*n - 1)!/(2*(n - 1)^2).

Sum_{n=2..oo} 1/a(n) = (e^2 - 3)/(4*e) = 0.40366087623617955676434290... . - Stefano Spezia, Jul 25 2024, simplified by Vaclav Kotesovec, Aug 19 2025

Sum_{n>=2} (-1)^n/a(n) = cos(1)/2. - Amiram Eldar, Aug 19 2025

Mathematica

Table[(2 n - 1)!/(2 (n - 1)^2), {n, 2, 20}]

Crossrefs

Cf. A001105, A009445, A007878, A138897, A138898.

Sequence in context: A013353 A270401 A270001 * A090627 A070234 A036279

Adjacent sequences: A138893 A138894 A138895 * A138897 A138898 A138899

Keyword

nonn

Author

Artur Jasinski, Apr 02 2008

Extensions

a(15)-a(17) from Stefano Spezia, Jul 25 2024

Status

approved