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Decimal expansion of Sum_{n>0} n/exp(n).
2

%I #28 Jul 22 2024 03:12:48

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

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

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

%N Decimal expansion of Sum_{n>0} n/exp(n).

%C The expression generating this constant is a first degree Eulerian polynomial, in the "variable" e, with coefficient {1}, generated from sum_{n>=0} n^m/e^n, with m=1. See A008292. It approximates m!. - _Richard R. Forberg_, Feb 15 2015

%C See A255169 for the second degree polynomial and value.

%F Equals exp(1)/(exp(1)-1)^2.

%F From _Gleb Koloskov_, Jul 12 2021: (Start)

%F Equals (1/2)/(cosh(1)-1).

%F Equals 1+Sum_{n>0} B(2*n)*(1-2*n)/(2*n)! = 1+Sum_{n>0} (A027641(2*n)/A027642(2*n))*A165747(n)/A010050(n).

%F Equals LambertW(x)*LambertW(-1,x), where x = (1/(1-e))*exp(1/(1-e)) = -A073333*exp(-A073333). (End)

%e 0.9206735942077923189454135227164996028816556266505511523539604097220...

%p g:=x->sum(n/exp(n),n=1..x); evalf[110](g(1500)); evalf[110](g(4000));

%t RealDigits[E/(E-1)^2, 10, 105][[1]] (* _Jean-François Alcover_, Jan 28 2014 *)

%o (PARI) 1+sumalt(n=1,bernreal(2*n)*(1-2*n)/(2*n)!) \\ _Gleb Koloskov_, Jul 12 2021

%Y Cf. A001113, A008292, A010050, A027641, A027642, A073333, A165747, A255169.

%K cons,nonn

%O 0,1

%A Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 03 2004