A036679 -id:A036679 - 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

A061711 a(n) = n^n * n!.

Original entry on oeis.org

1, 1, 8, 162, 6144, 375000, 33592320, 4150656720, 676457349120, 140587147048320, 36288000000000000, 11388728893445164800, 4270826380475341209600, 1886009588552176549862400, 968725766854884321342259200, 572622616354851562500000000000

Offset: 0

Lorenzo Fortunato (fortunat(AT)pd.infn.it), Jun 19 2001

a(n) is the product of first n terms of an arithmetic progression with first term n and common difference n. E.g. a(3) = 3*6*9 = 162. - Amarnath Murthy, Sep 20 2003

Product of the entries in the last column of an n X n square array whose elements are the numbers 1..n^2 listed in increasing order by rows. - Wesley Ivan Hurt, Mar 31 2025

			a(1) = 1^1 * 1! = 1;
a(2) = 2^2 * 2! = 8;
a(3) = 3^3 * 3! = 162.

Harry J. Smith, Table of n, a(n) for n = 0..100 (corrected by Michel Marcus, Jan 19 2019)

Cf. A036679, A053042, A055775. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009
Main diagonal of A131182.
Cf. A336765.

[Factorial(n)*n^n: n in [0..30]]; // G. C. Greubel, Nov 29 2022

Table[If[n == 0, 1, n^n] * n!, {n, 0, 20}] (* Vaclav Kotesovec, Mar 08 2018 *)

a(n) = n!*n^n; \\ Harry J. Smith, Jul 26 2009

from math import factorial
def A061711(n): return factorial(n)*n**n # Chai Wah Wu, Sep 03 2022

E.g.f.: sinh(n*x)^n. - Vaclav Kotesovec, Nov 05 2014
a(n) = [x^n] 1/(1 - n*x/(1 - n*x/(1 - 2*n*x/(1 - 2*n*x/(1 - 3*n*x/(1 - 3*n*x/(1 - ...))))))), a continued fraction. - Ilya Gutkovskiy, Sep 20 2017
Sum_{n>=1} 1/a(n) = A336765. - Amiram Eldar, Nov 20 2020
a(n) ~ exp(-n)*n^(2*n)*sqrt(2*n*Pi). - Peter Luschny, Jan 10 2022

A053042 a(n) = n^n + n!.

Original entry on oeis.org

2, 6, 33, 280, 3245, 47376, 828583, 16817536, 387783369, 10003628800, 285351587411, 8916579449856, 302881333613053, 11112094003849216, 437895198055227375, 18446764996499439616, 827240617573764860177, 39346414477670243303424

Offset: 1

N. J. A. Sloane, Feb 24 2000

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A000142, A000312, A036679.

lst={};Do[AppendTo[lst, n^n+n! ], {n, 3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 20 2008 *)
Table[n^n+n!,{n,20}] (* Harvey P. Dale, Dec 31 2016 *)

E.g.f.: 1/(1 + LambertW(-x)) + 1/(1 - x), where LambertW() is the Lambert W-function. - Ilya Gutkovskiy, Jan 08 2017

A101030 Triangle read by rows: T(n,k) = number of functions from an n-element set into but not onto a k-element set.

Original entry on oeis.org

0, 0, 2, 0, 2, 21, 0, 2, 45, 232, 0, 2, 93, 784, 3005, 0, 2, 189, 2536, 13825, 45936, 0, 2, 381, 7984, 61325, 264816, 818503, 0, 2, 765, 24712, 264625, 1488096, 5623681, 16736896, 0, 2, 1533, 75664, 1119005, 8172576, 38025127, 132766208, 387057609, 0

Offset: 1

Clark Kimberling, Nov 26 2004

			T(3,3) = #(functions into) - #(functions onto) = 3^3 - 6 = 21
Triangle T(n,k) begins:
  0,
  0, 2;
  0, 2,   21;
  0, 2,   45,   232;
  0, 2,   93,   784,    3005;
  0, 2,  189,  2536,   13825,   45936;
  0, 2,  381,  7984,   61325,  264816,   818503;
  0, 2,  765, 24712,  264625, 1488096,  5623681,  16736896;
  0, 2, 1533, 75664, 1119005, 8172576, 38025127, 132766208, 387057609;

Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.2(v).
D. P. Walsh, A note on non-surjective functions from [n] to [k].

Cf. A199656, A036679 (diagonal).

T:=(n, k)->sum((-1)^(j-1)*binomial(k, j)*(k-j)^n, j=1..k);
seq(seq(T(n, k), k=1..n), n=1..15); # Dennis P. Walsh, Apr 13 2016

T(n,k) = A089072(n,k) - A019538(n,k).
T(n,k) = Sum_{j=1..k} (-1)^(j-1)*C(k,j)*(k-j)^n. - Dennis P. Walsh, Apr 13 2016
T(n,k) = k^n - k!*Stirling2(n,k). - Dennis P. Walsh, Apr 13 2016

Offset corrected from 0 to 1 by Dennis P. Walsh, Apr 13 2016

A344112 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not functions.

Original entry on oeis.org

1, 3, 12, 7, 56, 485, 15, 240, 4015, 65280, 31, 992, 32525, 1047552, 33551307, 63, 4032, 261415, 16773120, 1073726199, 68719430080, 127, 16256, 2094965, 268419072, 34359660243, 4398046231168, 562949952597769, 255, 65280, 16770655, 4294901760, 1099511237151

Offset: 1

Mohammad K. Azarian, May 10 2021

			T(2,2) = (number of relations) - (number of functions) = 2^4 - 4 = 12.
Triangle T(n,k) begins:
   1;
   3,  12;
   7,  56,   485;
  15, 240,  4015,   65280;
  31, 992, 32525, 1047552, 33551307;

Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows n = 1..50, flattened)
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A101030, A199656, A036679, A344110.

Column[Table[2^(n*k) - k^n, {n, 10}, {k, n}], Center]

T(n,k) = 2^(n*k) - k^n, n,k >= 1.

A344113 a(n) = 2^(n^2) - n^n.

Original entry on oeis.org

1, 12, 485, 65280, 33551307, 68719430080, 562949952597769, 18446744073692774400, 2417851639229257961991863, 1267650600228229401486703205376, 2658455991569831745807613835249018541, 22300745198530623141535718272639445405532160

Offset: 1

Mohammad K. Azarian, May 14 2021

a(n) is the number of relations on a set with n elements that are not functions.

			a(1) = 2^(1^2) - 1^1 = 1.
a(2) = 2^(2^2) - 2^2 = 12.
a(3) = 2^(3^2) - 3^3 = 485.

Michael De Vlieger, Table of n, a(n) for n = 1..57
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A344110, A344112, A344114.

Table[2^(n^2) - n^n, {n, 12}] // Flatten

A344114 a(n) = 2^(n^2) - n!.

Original entry on oeis.org

1, 14, 506, 65512, 33554312, 68719476016, 562949953416272, 18446744073709511296, 2417851639229258349049472, 1267650600228229401496699576576, 2658455991569831745807614120520772352, 22300745198530623141535718272648361026978816, 748288838313422294120286634350736906063831234982912

Offset: 1

Mohammad K. Azarian, Jun 04 2021

a(n) is the number of relations on a set with n elements that are not one-to-one functions.

			a(1) = 2^(1^2) - 1! =   1;
a(2) = 2^(2^2) - 2! =  14;
a(3) = 2^(3^2) - 3! = 506.

Michael De Vlieger, Table of n, a(n) for n = 1..57
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A101030, A199656, A344110, A344112, A344113.

Mathematica
```
Table[2^(n^2) - n!, {n, 16}] // Flatten
```

A162603 Primes of the form k^k - k! + 1.

Original entry on oeis.org

3, 233, 9996371201, 11111919647266817

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 07 2009

nonn

The next term is too large to include.

The next term, a(5), has 527 digits and derives from n=224. - Harvey P. Dale, Jun 03 2014

			2^2-2!+1 = 4-2+1 = 3, 4^4-4!+1 = 256-24+1 = 233, ...

Primes of the form A036679(k)+1.

f[n_]:=n^n-n!+1; lst={};Do[p=f[n];If[PrimeQ[p],AppendTo[lst,p]],{n,2*5!}];lst
Select[Table[n^n-n!+1,{n,30}],PrimeQ] (* Harvey P. Dale, Jun 03 2014 *)

A344115 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not one-to-one functions.

Original entry on oeis.org

1, 2, 14, 5, 58, 506, 12, 244, 4072, 65512, 27, 1004, 32708, 1048456, 33554312, 58, 4066, 262024, 16776856, 1073741104, 68719476016, 121, 16342, 2096942, 268434616, 34359735848, 4398046506064, 562949953416272, 248, 65480, 16776880, 4294965616, 1099511621056

Offset: 1

Mohammad K. Azarian, Jun 06 2021

If n=k, then T(n,k) = 2^(n^2) - n!, which is A344114, and if kA344110.

			For T(2,2): the number of relations is 2^4 and the number of one-to-one functions is 2, so 2^4 - 2 = 14 and thus T(2,2) = 14.
Triangle T(n,k) begins:
   1;
   2,   14;
   5,   58,   506;
  12,  244,  4072,   65512;
  27, 1004, 32708, 1048456, 33554312;

Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows n = 1..50, flattened)
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A068424, A101030, A199656, A344110, A344112, A344113, A344114.

Table[2^(n*k) - k!/(k - n)!, {k, 10}, {n, k}] // Flatten

T(n,k) = 2^(n*k) - k!/(k-n)!, k >= n.

A057157 Number of non-invertible functions from {0,1}^n to {0,1}^n.

Original entry on oeis.org

0, 2, 232, 16736896, 18446723150919663616, 1461501637330639787366751139186115801643772542976

Offset: 0

Henry Bottomley, Aug 15 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..10

Cf. A057156, A000722, A036679, A000079.

[(2^n)^(2^n)-Factorial(2^n): n in [0..5]]; // Vincenzo Librandi, Aug 22 2011

Table[(2^n)^(2^n) - (2^n)!, {n,0,5}] (* G. C. Greubel, Nov 08 2018 *)

vector(6,n,n--; (2^n)^(2^n) - (2^n)!) \\ G. C. Greubel, Nov 08 2018

a(n) = (2^n)^(2^n) - (2^n)! = A057156(n) - A000722(n) = A036679(A000079(n)).

cp's OEIS Frontend

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

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A061711 a(n) = n^n * n!.

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A053042 a(n) = n^n + n!.

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A101030 Triangle read by rows: T(n,k) = number of functions from an n-element set into but not onto a k-element set.

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A344112 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not functions.

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A344113 a(n) = 2^(n^2) - n^n.

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A344114 a(n) = 2^(n^2) - n!.

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A162603 Primes of the form k^k - k! + 1.