A073225 -id:A073225 - cp's OEIS frontend

A055775 a(n) = floor(n^n / n!).

1, 1, 2, 4, 10, 26, 64, 163, 416, 1067, 2755, 7147, 18613, 48638, 127463, 334864, 881657, 2325750, 6145596, 16263866, 43099804, 114356611, 303761260, 807692034, 2149632061, 5726042115, 15264691107, 40722913454, 108713644516

Offset: 0

Henry Bottomley, Jul 12 2000

Stirling's approximation for n! suggests that this should be about e^n/sqrt(pi*2n). Bill Gosper has noted that e^n/sqrt(pi*(2n+1/3)) is significantly better.
n^n/n! = A001142(n)/A001142(n-1), where A001142(n) is product{k=0 to n} C(n,k) (where C() is a binomial coefficient). - Leroy Quet, May 01 2004
There are n^n distinct functions from [n] to [n] or sequences on n symbols of length n, the number of those sequences having n distinct symbols is n!. So the probability P(n) of bijection is n!/n^n. The expected value of the number of functions that we pick until we found a bijection is the reciprocal of P(n), or n^n/n!. - Washington Bomfim, Mar 05 2012

			a(5)=26 since 5^5=3125, 5!=120, 3125/120=26.0416666...

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Washington Bomfim, A method to find bijections from a set of n integers to {0,1, ... ,n-1}
Eric Weisstein's World of Mathematics, Stirling's Approximation for n!

Cf. A073225, A094082, A053042, A036679, A061711, A152170, A209081, A208846, A208847.

[Floor((n^n)/Factorial(n)): n in [0..30]]; // Vincenzo Librandi, Aug 22 2011

Join[{1}, Table[Floor[n^n/n!], {n, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)

a(n)=n^n\n! \\ Charles R Greathouse IV, Apr 17 2012

a(n) = floor(A000312(n)/A000142(n)).

More terms from James Sellers, Jul 13 2000

A094082 Decimal expansion of Sum_{n>=1} (n!/n^n).

Original entry on oeis.org

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

Offset: 1

Ross La Haye, May 01 2004

Sum_{n>=1} n!/n^(n+2) = Integral_{x=0..infinity} -log(1-x*exp(-x)) dx = 1.157694752682... . - Vaclav Kotesovec, Jan 05 2016

			1.879853862175258533486306145...

Cf. A055775, A073225.

RealDigits[ NSum[n!/n^n, {n, 1, Infinity}, WorkingPrecision -> 110, NSumTerms -> 180] , 10, 99] // First (* Jean-François Alcover, Feb 12 2013 *)
RealDigits @ NIntegrate[x E^x/(E^x - x)^2, {x, 0, Infinity}, WorkingPrecision -> 105] // First (* Michael Somos, May 18 2021 *)

firstDecimalDigits(n)={default(realprecision,n);return(digits(floor(suminf(n=1,n!/(n^n))*10^n)))}
print(firstDecimalDigits(98)); \\ R. J. Cano, Dec 30 2016

suminf(n=1,n!/(n^n)) \\ Michel Marcus, Dec 30 2016

Equals Integral_{x=0..oo} x*exp(x)/(exp(x)-x)^2 dx. - Michael Somos, May 18 2021

Equals Integral_{x=1..oo} 1/(x - log(x))^2 dx. - Fabián Pereyra, May 10 2023

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), May 01 2004

A235496 a(n) = round(n^n/n!) where round(1/2)=1.

Original entry on oeis.org

1, 1, 2, 5, 11, 26, 65, 163, 416, 1068, 2756, 7148, 18614, 48639, 127463, 334865, 881658, 2325751, 6145597, 16263866, 43099804, 114356611, 303761260, 807692035, 2149632061, 5726042115, 15264691107, 40722913454, 108713644516, 290404350963, 776207020880

Offset: 0

Vincenzo Librandi, Jan 15 2014

We have two versions of this sequence, this and A240571, because there is no universal agreement on how to round a number like 9/2. - N. J. A. Sloane, Dec 13 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000142, A000312, A055775, A073225.

See A240571 for another version.

[Round(n^n/Factorial(n)): n in [1..40]];

a:= n-> round(n^n/n!):
seq(a(n), n=0..32);  # Alois P. Heinz, Dec 13 2015

Mathematica
```
Table[Floor[n^n/n! + 1/2], {n, 40}]
```

s=[]; for(n=1, 30, s=concat(s, round(n^n/n!))); s \\ Colin Barker, Jan 19 2014

a(n) = Round(A000312(n)/A000142(n)).

Name clarified by Sean A. Irvine, Jan 12 2025

A127634 a(n) = 3^(n-1) - ceiling(n^n/n!).

Original entry on oeis.org

0, 1, 4, 16, 54, 178, 565, 1770, 5493, 16927, 51901, 158533, 482802, 1466859, 4448104, 13467249, 40720970, 122994566, 371156622, 1119161662, 3372427789, 10156591942, 30573367574, 91993546765, 276703494365, 832023918335, 2501142914874, 7516883840470

Offset: 1

N. J. A. Sloane, Apr 03 2007

nonn

Theorem: 3^(n-1) > n^n/n! for n >= 3.

D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970; p. 193, 3.1.21.

Robert Israel, Table of n, a(n) for n = 1..2094

Cf. A000244, A073225.

[3^(n-1)-Ceiling(n^n/Factorial(n)): n in [1..30]]; // Vincenzo Librandi, Jul 06 2017

seq(3^(n-1)-ceil(n^n/n!),n=1..50); # Robert Israel, Jul 06 2017

Table[3^(n-1) - Ceiling[n^n / n!], {n, 30}] (* Vincenzo Librandi, Jul 06 2017 *)

a(n) = 3^(n-1) - ceil(n^n/n!); \\ Michel Marcus, Jul 06 2017

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A055775 a(n) = floor(n^n / n!).

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A094082 Decimal expansion of Sum_{n>=1} (n!/n^n).

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A235496 a(n) = round(n^n/n!) where round(1/2)=1.

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A127634 a(n) = 3^(n-1) - ceiling(n^n/n!).

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