A248682 Decimal expansion of Sum_{n >= 0} (floor(n/2)!)^2/n!.

2, 9, 4, 5, 5, 9, 9, 4, 3, 4, 8, 7, 4, 8, 6, 0, 3, 1, 1, 6, 3, 9, 1, 8, 0, 6, 7, 3, 4, 5, 9, 6, 9, 3, 9, 8, 4, 2, 5, 2, 5, 0, 3, 3, 3, 1, 6, 3, 7, 9, 9, 1, 6, 2, 2, 7, 2, 8, 7, 8, 6, 6, 0, 9, 2, 3, 3, 8, 8, 7, 2, 7, 2, 1, 1, 2, 3, 1, 4, 5, 6, 3, 2, 7, 4, 7

Offset: 1

Author

Clark Kimberling, Oct 11 2014

Keywords

• nonn
• easy
• cons

Comments

Limit_{x -> inf} Sum {n=0..inf} (Floor[n/x])!^x/n! = e (A001113).

For A248682: x = 2; A248683: x = 3; A248684: x = 4; A248685: x = 5. - Robert G. Wilson v, Feb 22 2016

Let n} denote the swinging factorial A056040(n), then the constant equals Sum_{n>=0} 1/n} and is sometimes called the swinging constant e}. ("e}" is written in TeX $e\wr$). For a proof that it equals 3^(1/2)*(2/3)^3*Pi + 4/3 see the link to Mathematics Stack Exchange. - Peter Luschny, Jul 22 2022

Examples

			2.94559943487486031163918067345969398425250...
		

Links

• Georg Fischer, Table of n, a(n) for n = 1..1000 (first 149 terms from Clark Kimberling)
• Jan Eerland, Answer to question 3689772, Mathematics Stack Exchange, 2021. See also question 3692793.
• Index entries for transcendental numbers

Crossrefs

Cf. A001113, A248683, A248684, A248785, A248664, A056040 (swinging factorial).

Programs

Mathematica

RealDigits[Sum[(Floor[n/2])!^2/n!, {n, 0, 400}], 10, 111][[1]]
RealDigits[4/3+8Pi/Sqrt[243],10,111][[1]] (* Robert G. Wilson v, Feb 10 2016 *)

PARI

suminf(n=0, ((n\2)!)^2/n!) \\ Michel Marcus, Feb 11 2016

Formula

Equals Sum_{n >= 0} (n!^2)*p(2,n)/(2*n + 1)!, where p(k,n) is defined at A248664.

Equals Sum_{n >= 0} (floor(n/2)!)^2/n! = Sum_(n >= 1) (3n^2 - 7n + 6)/C(2n, n) = 4/3 + 8*Pi/sqrt(243). - Robert G. Wilson v, Feb 11 2016

Equals 1 + Integral_{x>=0} 1/(x^2 - x + 1)^2 dx. - Amiram Eldar, Nov 16 2021