A055775 -id:A055775 - cp's OEIS frontend

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

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

A090582 T(n, k) = Sum_{j=0..n-k} (-1)^jbinomial(n - k + 1, j)(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 24, 36, 14, 1, 120, 240, 150, 30, 1, 720, 1800, 1560, 540, 62, 1, 5040, 15120, 16800, 8400, 1806, 126, 1, 40320, 141120, 191520, 126000, 40824, 5796, 254, 1, 362880, 1451520, 2328480, 1905120, 834120, 186480, 18150, 510, 1, 3628800, 16329600, 30240000, 29635200, 16435440, 5103000, 818520, 55980, 1022, 1

Offset: 1

Hugo Pfoertner, Jan 11 2004

Let Q(m, n) = Sum_(k=0..n-1) (-1)^k * binomial(n, k) * (n-k)^m. Then Q(m,n) is the numerator of the probability P(m,n) = Q(m,n)/n^m of seeing each card at least once if m >= n cards are drawn with replacement from a deck of n cards, written in a two-dimensional array read by antidiagonals with Q(m,m) as first row and Q(m,1) as first column.
The sequence is given as a matrix with the first row containing the cases #draws = size_of_deck. The second row contains #draws = 1 + size_of_deck. If "mn" indicates m cards drawn from a deck with n cards then the locations in the matrix are:
  11 22 33 44 55 66 77 ...
  21 32 43 54 65 76 87 ...
  31 42 53 64 75 86 97 ...
  41 52 63 74 85 .. .. ...
read by antidiagonals ->:
  11, 22, 21, 33, 32, 31, 44, 43, 42, 41, 55, 54, 53, 52, ....
The probabilities are given by Q(m,n)/n^m:
.(m,n):.....11..22..21..33..32..31..44..43..42..41...55...54..53..52..51
.....Q:......1...2...1...6...6...1..24..36..14...1..120..240.150..30...1
...n^m:......1...4...1..27...8...1.256..81..16...1.3125.1024.243..32...1
%.Success:.100..50.100..22..75.100...9..44..88.100....4...23..62..94.100
P(n,n) = n!/n^n which can be approximated by sqrt(Pi*(2n+1/3))/e^n (Gosper's approximation to n!).
Let P[n] be the set of all n-permutations. Build a superset Q[n] of P[n] composed of n-permutations in which some (possibly all or none) ascents have been designated. An ascent in a permutation s[1]s[2]...s[n] is a pair of consecutive elements s[i],s[i+1] such that s[i] < s[i+1]. As a triangular array read by rows T(n,k) is the number of elements in Q[n] that have exactly k distinguished ascents, n >= 1, 0 <= k <= n-1. Row sums are A000670. E.g.f. is y/(1+y-exp(y*x)). For example, T(3,1)=6 because there are four 3-permutations with one ascent, with these we would also count 1->2 3, and 1 2->3 where exactly one ascent is designated by "->". (After Flajolet and Sedgewick.) - Geoffrey Critzer, Nov 13 2012
Sum_(k=1..n) Q(n, k)*binomial(x, k) = x^n such that Sum_{k=1..n} Q(n, i)*binomial(x+1,i+1) = Sum_{k=1..x} k^n. - David A. Corneth, Feb 17 2014
A141618(n,n-k+1) = a(n,k) * C(n,k-1) / k. - Tom Copeland, Oct 25 2014
See A074909 and above g.f.s below for associations among this array and the Bernoulli polynomials and their umbral compositional inverses. - Tom Copeland, Nov 14 2014
For connections to toric varieties and Eulerian polynomials (in addition to those noted below), see the Dolgachev and Lunts and the Stembridge links in A019538. - Tom Copeland, Dec 31 2015
See A008279 for a relation between the e.g.f.s enumerating the faces of permutahedra (this entry) and stellahedra. - Tom Copeland, Nov 14 2016
From the Hasan and Franco and Hasan papers: The m-permutohedra for m=1,2,3,4 are the line segment, hexagon, truncated octahedron and omnitruncated 5-cell. The first three are well-known from the study of elliptic models, brane tilings and brane brick models. The m+1 torus can be tiled by a single (m+2)-permutohedron. Relations to toric Calabi-Yau Kahler manifolds are also discussed. - Tom Copeland, May 14 2020

			For m = 4, n = 2, we draw 4 times from a deck of two cards. Call the cards "a" and "b" - of the 16 possible combinations of draws (each of which is equally likely to occur), only two do not contain both a and b: a, a, a, a and b, b, b, b. So the probability of seeing both a and b is 14/16. Therefore Q(m, n) = 14.
Table starts:
  [1] 1;
  [2] 2,      1;
  [3] 6,      6,       1;
  [4] 24,     36,      14,      1;
  [5] 120,    240,     150,     30,      1;
  [6] 720,    1800,    1560,    540,     62,     1;
  [7] 5040,   15120,   16800,   8400,    1806,   126,    1;
  [8] 40320,  141120,  191520,  126000,  40824,  5796,   254,   1;
  [9] 362880, 1451520, 2328480, 1905120, 834120, 186480, 18150, 510, 1.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Kevin Buhr, Michael Press and DMB, Collecting a deck of cards on the street. Thread in NG sci.math.
José L. Cereceda, Figurate numbers and sums of powers of integers, arXiv:2001.03208 [math.NT], 2020. See triangle Table 1 p. 5.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 209.
S. Franco and A. Hasan, Graded Quivers, Generalized Dimer Models and Toric Geometry, arXiv preprint arXiv:1904.07954 [hep-th], 2019.
A. Hasan, Physics and Mathematics of Graded Quivers, dissertation, Graduate Center, City University of New York, 2019.
Hugo Pfoertner, "Hit all" probabilities: Program and results.

Cf. A073593 first m >= n giving at least 50% probability, A085813 ditto for 95%, A055775 n^n/n!, A090583 Gosper's approximation to n!.
Reflected version of A019538.
Cf. A233734 (central terms).
Cf. A074909, A008292, A028246.
Cf. A046802, A119879, A123125, A130850, and A248727.

a090582 n k = a090582_tabl !! (n-1) !! (k-1)
a090582_row n = a090582_tabl !! (n-1)
a090582_tabl = map reverse a019538_tabl
-- Reinhard Zumkeller, Dec 15 2013

T := (n, k) -> add((-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n, j = 0..n-k):
# Or:
T := (n, k) -> (n - k + 1)!*Stirling2(n, n - k + 1):
for n from 1 to 9 do seq( T(n, k), k = 1..n) od; # Peter Luschny, May 21 2021

In[1]:= Table[Table[k! StirlingS2[n, k], {k, n, 1, -1}], {n, 1, 6}] (* Victor Adamchik, Oct 05 2005 *)
nn=6; a=y/(1+y-Exp[y x]); Range[0,nn]! CoefficientList[Series[a, {x,0,nn}], {x,y}]//Grid (* Geoffrey Critzer, Nov 10 2012 *)

a(n)={m=ceil((-1+sqrt(1+8*n))/2);k=m+1+binomial(m,2)-n;k*sum(i=1,k,(-1)^(i+k)*i^(m-1)*binomial(k-1,i-1))} \\ David A. Corneth, Feb 17 2014

T(n, k) = (n - k + 1)!*Stirling2(n, n - k + 1), generated by Stirling numbers of the second kind multiplied by a factorial. - Victor Adamchik, Oct 05 2005
Triangle T(n,k), 1 <= k <= n, read by rows given by [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] DELTA [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2006
From Tom Copeland, Oct 07 2008: (Start)
G(x,t) = 1/ (1 + (1-exp(x*t))/t) = 1 + 1*x + (2 + t)*x^2/2! + (6 + 6*t + t^2)*x^3/3! + ... gives row polynomials of A090582, the f-polynomials for the permutohedra (see A019538).
G(x,t-1) = 1 + 1*x + (1 + t)*x^2/2! + (1 + 4*t + t^2)*x^3/3! + ... gives row polynomials for A008292, the h-polynomials of permutohedra.
G[(t+1)x,-1/(t+1)] = 1 + (1 + t)*x + (1 + 3*t + 2*t^2)*x^2/2! + ... gives row polynomials of A028246. (End)
From Tom Copeland, Oct 11 2011: (Start)
With e.g.f. A(x,t) = G(x,t) - 1, the compositional inverse in x is
B(x,t) = log((t+1)-t/(1+x))/t. Let h(x,t) = 1/(dB/dx) = (1+x)*(1+(1+t)x), then the row polynomial P(n,t) is given by (1/n!)*(h(x,t)*d/dx)^n x, evaluated at x=0, A=exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t). (End)
k <= 0 or k > n yields Q(n, k) = 0; Q(1,1) = 1; Q(n, k) = k * (Q(n-1, k) + Q(n-1, k-1)). - David A. Corneth, Feb 17 2014
T = A008292*A007318. - Tom Copeland, Nov 13 2016
With all offsets 0 for this entry, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125 with offsets -1 so that the array becomes A008292; i.e., we ignore the first row and first column of A123125. Then the row polynomials of this entry, A090582, are given by A_n(1 + x;0). Other specializations of A_n(x;y) give A028246, A046802, A119879, A130850, and A248727. - Tom Copeland, Jan 24 2020

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

A073225 a(n) = ceiling(n^n/n!).

Original entry on oeis.org

1, 1, 2, 5, 11, 27, 65, 164, 417, 1068, 2756, 7148, 18614, 48639, 127464, 334865, 881658, 2325751, 6145597, 16263867, 43099805, 114356612, 303761261, 807692035, 2149632062, 5726042116, 15264691108, 40722913455, 108713644517

Offset: 0

Michael Somos, Jul 22 2002

nonn

The van der Waerden conjecture, now a theorem thanks to Egorycev, states that the permanent of any n X n doubly stochastic matrix is >= n!/n^n, with equality iff the matrix has all entries equal to 1/n.
Therefore the reciprocal of the permanent of any n X n doubly stochastic matrix is bounded from above by n^n/n! and this sequence.
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

			G.f.: 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 27*x^5 + 65*x^6 + 164*x^7 + 417*x^8 + ...

G. P. Egorycev, Solution of the van der Waerden problem for permanents (Russian), Preprint IFSO-13 M. Akad. Nauk SSSR Sibirsk. Otdel., Inst. Fiz., Krasnoyarsk, 1980. 12 pp. Math. Rev. 82e:15006.
J. H. van Lint, R. M. Wilson, A Course in Combinatorics, Cambridge Univ. Press, 1992. p. 86.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A055775, A094082.

[Ceiling(n^n/Factorial(n)): n in [0..50]]; // G. C. Greubel, May 29 2018

Join[{1}, Table[Ceiling[n^n/n!], {n,1,50}]] (* G. C. Greubel, May 29 2018 *)

PARI
```
{a(n) = ceil(n^n / n!)}
```

A090583 Gosper's approximation to n!, sqrt((2n+1/3)Pi)*n^n/e^n, rounded to nearest integer.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5039, 40316, 362851, 3628561, 39914615, 478979481, 6226774954, 87175314872, 1307635379670, 20922240412500, 355679137660826, 6402240370021199, 121642823201649058, 2432860847996122437, 51090157192742729183, 1123984974735953018069

Offset: 0

Hugo Pfoertner, Jan 10 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 0..449 (terms 0..200 from Alois P. Heinz)
Peter Luschny, Approximations to the Factorial Function.
Eric Weisstein's World of Mathematics, Stirling's Approximation.

Cf. A000142, A005394, A055775.

C := ComplexField(); [Round(Sqrt((2*n + 1/3)*Pi(C))*n^n/Exp(n)): n in [0..30]]; // G. C. Greubel, Nov 28 2017

Maple

Digits:= 2000; a:= n-> round(sqrt((2*n+1/3)*Pi)*n^n/exp(n)): seq(a(n), n=0..30); # Alois P. Heinz, Feb 04 2013

Mathematica

Join[{1}, Table[Round[Sqrt[(2*n + 1/3)*Pi]*n^n/Exp[n]], {n, 1, 50}]] (* G. C. Greubel, Nov 28 2017 *)

PARI

a(n) = round(sqrt((2*n+1/3)*Pi)*n^n/exp(n)); \\ Bill McEachen, Aug 16 2014

A208846 a(n) = A056915(n) mod 76057 mod 13.

Original entry on oeis.org

2, 7, 11, 10, 5, 9, 6, 3, 4, 0, 1, 12, 8, 8, 6, 10, 9, 6, 7, 10, 3, 6, 2, 9, 8, 1, 2, 2, 2, 8, 0, 5, 5, 2, 7, 11, 5, 2, 11, 0, 10, 8, 2, 7, 4, 10, 2, 0, 5, 12, 8, 11, 6, 7, 7, 11, 0, 5, 1, 12, 6, 4, 6, 7, 8, 1, 12, 0, 7, 2, 9

Offset: 1

Washington Bomfim, Mar 02 2012

nonn

A056915(n) mod 76057 mod 13 is a bijection from the set of the first 13 terms of A056915 to {0,1,2,3,4,5,6,7,8,9,10,11,12}.
One of the tests for primality described in the first reference when tests x and x is prime, searches a table T composed by the first 13 entries of A056915 to see if x is a strong pseudoprime to bases 2,3 and 5. A fast way to do that is to compute i = x  mod 76057 mod 13, and compare x with T[i]. If x is not equal to T[i], x is prime.
Terms computed using table by Charles R Greathouse IV. See A056915.

Washington Bomfim, A method to find bijections from a set of n integers to {0,1, ... ,n-1}
C. Pomerance, J. L. Selfridge, and S. S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Mathematics of Computation, 35, 1980, pp. 1003-1026.

Cf. A055775.

A210112 Floor of the expected value of number of trials until exactly one cell is empty in a random distribution of n balls in n cells.

Original entry on oeis.org

2, 1, 1, 2, 4, 7, 14, 29, 61, 129, 282, 623, 1400, 3189, 7347, 17101, 40167, 95110, 226841, 544555, 1314983, 3192458, 7788521, 19086807, 46968280, 116019696, 287602234, 715281652, 1784383956, 4464139806

Offset: 2

Washington Bomfim, Mar 18 2012

nonn

Also floor of the expected value of number of trials until we have n-1 distinct symbols in a random sequence on n symbols of length n. A055775 corresponds to zero cells empty.

			For n=2, with symbols 0 and 1, the 2^2 sequences on 2 symbols of length 2 can be represented by 00, 01, 10, and 11. We have 2 sequences with a unique symbol, so a(2) = floor(4/2) = 2.

W. Feller, An Introduction to Probability Theory and its Applications, 2nd ed, Wiley, New York, 1965, (2.4) p. 92. (Occupancy problems)

Washington Bomfim, Table of n, a(n) for n = 2..100

Cf. A055775, A209899, A209900, A210113, A210114, A210115, A210116.

With m = 1, a(n) = floor(n^n/(binomial(n,m)_Sum{v=0..n-m-1}((-1)^v*binomial(n-m,v) (n-m-v)^n)))

A210113 Floor of the expected value of number of trials until exactly two cells are empty in a random distribution of n balls in n cells.

Original entry on oeis.org

9, 3, 2, 1, 2, 3, 4, 7, 12, 21, 40, 75, 147, 292, 594, 1229, 2582, 5499, 11859, 25868, 57008, 126814, 284523, 643401, 1465511, 3360493, 7753730, 17993787, 41982506, 98445184, 231932762, 548839352, 1304155087

Offset: 3

Washington Bomfim, Mar 18 2012

nonn

Also floor of the expected value of number of trials until we have n-2 distinct symbols in a random sequence on n symbols of length n. A055775 corresponds to zero cells empty.

			For n=3, there are 3^3 = 27 sequences on 3 symbols of length 3. Only 3 sequences has a unique symbol, so a(3) = floor(27/3) = 9.

W. Feller, An Introduction to Probability Theory and its Applications, 2nd ed, Wiley, New York, 1965, (2.4) p. 92. (Occupancy problems)

Washington Bomfim, Table of n, a(n) for n = 3..100

Cf. A055775, A209899, A209900, A210112, A210114, A210115, A210116.

With m = 2, a(n) = floor(n^n/(binomial(n,m)*_Sum{v=0..n-m-1}((-1)^v*binomial(n-m,v)*(n-m-v)^n)))

A210114 Floor of the expected value of number of trials until exactly three cells are empty in a random distribution of n balls in n cells.

Original entry on oeis.org

64, 10, 4, 2, 2, 2, 2, 3, 5, 7, 11, 18, 31, 55, 100, 185, 348, 670, 1311, 2606, 5254, 10734, 22196, 46407, 98023, 209009, 449580, 974963, 2130442, 4688533, 10387113, 23156162, 51926745, 117090391, 265413053

Offset: 4

Washington Bomfim, Mar 18 2012

nonn

Also floor of the expected value of number of trials until we have n-3 distinct symbols in a random sequence on n symbols of length n. A055775 corresponds to zero cells empty.

			For n=4, there are 4^4 = 256 sequences on 4 symbols of length 4. Only 4 sequences have a unique symbol, so a(4) = floor(256/4) = 64.

W. Feller, An Introduction to Probability Theory and its Applications, 2nd ed, Wiley, New York, 1965, (2.4) p. 92. (Occupancy problems)

Washington Bomfim, Table of n, a(n) for n = 4..100

Cf. A055775, A209899, A209900, A210112, A210113, A210115, A210116.

With m = 3, a(n) = floor(n^n/(binomial(n,m)*_Sum{v=0..n-m-1}((-1)^v*binomial(n-m,v)*(n-m-v)^n)))

A210115 Floor of the expected value of number of trials until exactly four cells are empty in a random distribution of n balls in n cells.

Original entry on oeis.org

625, 50, 13, 5, 3, 2, 2, 2, 3, 4, 5, 7, 11, 17, 28, 46, 78, 136, 242, 441, 815, 1533, 2927, 5669, 11123, 22090, 44363, 90027, 184482, 381499, 795686, 1672914, 3543925, 7561129, 16240832, 35106812, 76346759, 166982782, 367206632, 811693449

Offset: 5

Washington Bomfim, Mar 18 2012

nonn

Also floor of the expected value of number of trials until we have n-4 distinct symbols in a random sequence on n symbols of length n. A055775 corresponds to zero cells empty.

			For n=5, there are 5^5 = 3125 sequences on 5 symbols of length 5. Only 5 sequences has a unique symbol, so a(5) = floor(3125/5) = 625.

W. Feller, An Introduction to Probability Theory and its Applications, 2nd ed, Wiley, New York, 1965, (2.4) p. 92. (Occupancy problems)

W. Bomfim, Table of n, a(n) for n = 5..100

Cf. A055775, A209899, A209900, A210112, A210113, A210114, A210116.

With m = 4, a(n) = floor(n^n/(binomial(n,m)*_Sum{v=0..n-m-1}((-1)^v*binomial(n-m,v)*(n-m-v)^n)))

cp's OEIS Frontend

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

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A090582 T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.

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

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A073225 a(n) = ceiling(n^n/n!).

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A090583 Gosper's approximation to n!, sqrt((2*n+1/3)*Pi)*n^n/e^n, rounded to nearest integer.

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Maple

Mathematica

PARI

A208846 a(n) = A056915(n) mod 76057 mod 13.

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A210112 Floor of the expected value of number of trials until exactly one cell is empty in a random distribution of n balls in n cells.

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A090582 T(n, k) = Sum_{j=0..n-k} (-1)^jbinomial(n - k + 1, j)(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.

A090583 Gosper's approximation to n!, sqrt((2n+1/3)Pi)*n^n/e^n, rounded to nearest integer.