

A248682


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


9



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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

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


LINKS



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


EXAMPLE

2.94559943487486031163918067345969398425250...


MATHEMATICA

RealDigits[Sum[(Floor[n/2])!^2/n!, {n, 0, 400}], 10, 111][[1]]


PROG



CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



