A293656 a(n) = binomial(n+1,2)*n!/n!!.

1, 3, 12, 30, 120, 315, 1344, 3780, 17280, 51975, 253440, 810810, 4193280, 14189175, 77414400, 275675400, 1579253760, 5892561675, 35300966400, 137493105750, 858370867200, 3478575575475, 22562891366400, 94870242967500, 637646929920000, 2774954606799375

Offset: 1

Author

Justin Gaetano, Feb 06 2018

Keywords

• nonn

Comments

It appears that the difference between a(n)/a(n-1) and a(n-1)/a(n-2) approaches some factor, 5 < x < 7, as n --> oo.

It appears that 3|a(n) for n > 1.

Examples

			For n = 6, a(6) = binomial(6+1,2)/(6!!/6!) = 315.
		

References

• L. Euler and J. L. Lagrange, Elements of Algebra, J. Johnson and Co. 1810. See pages 332-335.

Crossrefs

Cf. A000217, A000142, A006882.

Programs

Mathematica

Table[Binomial[n+1,2] n!/n!!,{n,30}] (* Harvey P. Dale, Jul 27 2021 *)

Formula

a(n) = ((n*(n+1))/2)/(Product_{i=0..floor((n-1)/2),n-2*i}/Product_{i=1..n}).

From Chai Wah Wu, Feb 07 2018: (Start)

a(n) = n*(n+1)!!/2.

a(n)/a(n-1) = ((n+1)!!/n!!)*(n/(n-1)) = n/b*(n-1) if n is even and n*Pi/(2*b*(n-1)) if n is odd where b = Integral_{x=0..(Pi/2)} sin^(n+1)*x dx.

Since b -> 0 as n -> oo, a(n)/a(n-1) is unbounded as n -> oo. On the other hand, a(n)/a(n-1) and a(n-1)/a(n-2) differ by a multiplicative factor of approximately Pi/2.

(End)