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Ratio of (2n-1)! to number of zeros in Sylvester matrix of polynomial of n degree with all nonzero coefficients.
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%I #25 Jul 28 2024 00:15:58

%S 3,15,280,11340,798336,86486400,13343616000,2778808032000,

%T 750895681536000,255454710858547200,106826515449937920000,

%U 53858368206010368000000,32215590089995124736000000,22555515290152300904448000000,18272974787062050706056806400000,16959604724241965811558973440000000

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

%C (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.

%C 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

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

%F Sum_{n=2..oo} 1/a(n) = (1 - 3*e^2 + 8*e*sinh(1))/(4*e) = 0.40366087623617955676434290... . - _Stefano Spezia_, Jul 25 2024

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

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

%K nonn

%O 2,1

%A _Artur Jasinski_, Apr 02 2008

%E a(15)-a(17) from _Stefano Spezia_, Jul 25 2024