A318910 a(n) is the nearest integer to binomial(n,n/2) = n!/((n/2)!)^2.

1, 1, 2, 3, 6, 11, 20, 37, 70, 132, 252, 482, 924, 1778, 3432, 6639, 12870, 24994, 48620, 94716, 184756, 360821, 705432, 1380533, 2704156, 5301248, 10400600, 20419624, 40116600, 78861995, 155117520, 305272239, 601080390, 1184086260, 2333606220, 4601020897, 9075135300

Offset: 0

Author

Jianing Song, Sep 05 2018

Keywords

• nonn

Comments

If we consider binomial(n,x) as a real-valued function of x, then a(n) is the maximum value of binomial(n,x) (always obtained by x = n/2), rounded. Here binomial(n,x) must be understood as Gamma(n+1)/(Gamma(x+1)*Gamma(n-x+1)).

A093581 actually has already mentioned a geometric interpretation of this sequence.

Examples

			a(1) = round(1!/(0.5!)^2) = round(4/Pi) = round(1.2732395...) = 1.
a(3) = round(3!/(1.5!)^2) = round(32/(3*Pi)) = round(3.3953054...) = 3.
a(5) = round(5!/(2.5!)^2) = round(512/(15*Pi)) = round(10.8649774...) = 11.
a(7) = round(7!/(3.5!)^2) = round(4096/(35*Pi)) = round(37.2513512...) = 37.
		

Crossrefs

Cf. A000984, A001803, A002457, A056040, A093581.

Programs

PARI

a(n) = round(gamma(n+1)/gamma(n/2+1)^2)

Formula

a(2n) = A000984(n), a(2n+1) = round((2^(4n+2)*(n!)^2)/(Pi*(2n+1)!)) = round(2^(4n+2)/(Pi*A002457(n))) = round(A093581(n)/(Pi*A001803(n))).

a(n) ~ 2^n/sqrt(n*Pi/2).