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A248682 Decimal expansion of Sum_{n >= 0} (floor(n/2)!)^2/n!. 9

%I #37 Dec 09 2023 07:08:12

%S 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,

%T 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,

%U 8,7,2,7,2,1,1,2,3,1,4,5,6,3,2,7,4,7

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

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

%C For A248682: x = 2; A248683: x = 3; A248684: x = 4; A248685: x = 5. - _Robert G. Wilson v_, Feb 22 2016

%C 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

%H Clark Kimberling, <a href="/A248682/b248682.txt">Table of n, a(n) for n = 1..1000</a>

%H Jan Eerland, <a href="https://math.stackexchange.com/a/3689772">Answer to question 3689772</a>, Mathematics Stack Exchange, 2021. See also <a href="https://math.stackexchange.com/q/3692793">question 3692793</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

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

%F 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

%F Equals 1 + Integral_{x>=0} 1/(x^2 - x + 1)^2 dx. - _Amiram Eldar_, Nov 16 2021

%e 2.94559943487486031163918067345969398425250...

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

%t RealDigits[4/3+8Pi/Sqrt[243],10,111][[1]] (* _Robert G. Wilson v_, Feb 10 2016 *)

%o (PARI) suminf(n=0, ((n\2)!)^2/n!) \\ _Michel Marcus_, Feb 11 2016

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

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Oct 11 2014

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