A061711 -id:A061711 - 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

A303489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] 1/(1 - k*x)^(n/k).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 8, 60, 1, 1, 10, 105, 840, 1, 1, 12, 162, 1920, 15120, 1, 1, 14, 231, 3640, 45045, 332640, 1, 1, 16, 312, 6144, 104720, 1290240, 8648640, 1, 1, 18, 405, 9576, 208845, 3674160, 43648605, 259459200, 1, 1, 20, 510, 14080, 375000, 8648640, 152152000, 1703116800, 8821612800

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

			Square array begins:
      1,      1,       1,       1,       1,       1,  ...
      1,      1,       1,       1,       1,       1,  ...
      6,      8,      10,      12,      14,      16,  ...
     60,    105,     162,     231,     312,     405,  ...
    840,   1920,    3640,    6144,    9576,   14080,  ...
  15120,  45045,  104720,  208845,  375000,  623645,  ...
=========================================================
A(1,1) = 1;
A(2,1) = 2*3 = 6;
A(3,1) = 3*4*5 = 60;
A(4,1) = 4*5*6*7 = 840;
A(5,1) = 5*6*7*8*9 = 15120, etc.
...
A(1,2) = 1;
A(2,2) = 2*4 = 8;
A(3,2) = 3*5*7 = 105;
A(4,2) = 4*6*8*10 = 1920;
A(5,2) = 5*7*9*11*13 = 45045, etc.
...
A(1,3) = 1;
A(2,3) = 2*5 = 10;
A(3,3) = 3*6*9 = 162;
A(4,3) = 4*7*10*13 = 3640;
A(5,3) = 5*8*11*14*17 = 104720, etc.
...

Index entries for sequences related to factorial numbers

Columns k=1..5 give A000407, A113551, A303486, A303487, A303488.
Main diagonal gives A061711.
Cf. A008279, A131182, A256268, A265609.

Table[Function[k, n! SeriesCoefficient[1/(1 - k x)^(n/k), {x, 0, n}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, Product[k i + n, {i, 0, n - 1}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, k^n Pochhammer[n/k, n]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten

A(n,k) = Product_{j=0..n-1} (k*j + n).

A291699 a(n) = n^n(2n)!/(n!*(n + 1)!).

Original entry on oeis.org

1, 1, 8, 135, 3584, 131250, 6158592, 353299947, 23991418880, 1883638417518, 167960000000000, 16772331868538246, 1854655886442627072, 225005916687384753700, 29718395534545380311040, 4245313393689422607421875, 652233889532678001886494720, 107247390031799133661006687830

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..322

Main diagonal of A290605.

Cf. A000108, A000312, A001761, A061711.

seq(n^n*(2*n)!/n!/(n+1)!, n=0..50); # Robert Israel, Aug 30 2017

Join[{1}, Table[n^n (2 n)!/(n! (n + 1)!), {n, 1, 17}]]
Table[SeriesCoefficient[2/(1 + Sqrt[1 - 4 n x]), {x, 0, n}], {n, 0, 17}]
Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-n x, 1, {i, 1, n}]), {x, 0, n}], {n, 0, 17}]

a(n)=binomial(2*n,n)/(n+1)*n^n \\ Charles R Greathouse IV, Oct 23 2023

a(n) = [x^n] 2/(1 + sqrt(1 - 4*n*x)).
a(n) = [x^n] 1/(1 - n*x/(1 - n*x/(1 - n*x/(1 - n*x/(1 - n*x/(1 - ...)))))), a continued fraction.
a(n) = n! * [x^n] (BesselI(0,2*n*x) - BesselI(1,2*n*x))*exp(2*n*x).
a(n) = n^n*binomial(2*n,n)/(n + 1).
a(n) = A000312(n)*A000108(n).
a(n) = A290605(n,n).
a(n) ~ 4^n*n^(n-3/2)/sqrt(Pi).

A152684 a(n) is the number of top-down sequences (F_1, F_2, ..., F_n) whereas each F_i is a labeled forest on n nodes, containing i directed rooted trees. F_(i+1) is proper subset of F_i.

Original entry on oeis.org

1, 2, 18, 384, 15000, 933120, 84707280, 10569646080, 1735643790720, 362880000000000, 94121726392108800, 29658516531078758400, 11159820050604594969600, 4942478402320838374195200, 2544989406021562500000000000, 1507645899890367707813511168000

Offset: 1

Fabian Nedic, Dec 10 2008

nonn

			a(1) = 1^(1-2)*(1!) = 1.
a(2) = 2^(2-2)*(2!) = 2.
a(3) = 3^(3-2)*(3!) = 18.

Miklos Bona, Introduction to Enumerative Combinatorics, McGraw Hill 2007, Page 276.

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A061711, A066319, A137268.

[Factorial(n-1)*n^(n-1): n in [1..20]]; // G. C. Greubel, Nov 28 2022

a:= proc(n) option remember; `if`(n=1, 1,
      a(n-1)*(n/(n-1))^(n-3)*n^2)
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, May 16 2013

Table[n^(n - 1) (n - 1)!, {n, 1, 16}]  (* Geoffrey Critzer, May 10 2013 *)

[factorial(n-1)*n^(n-1) for n in range(1,21)] # G. C. Greubel, Nov 28 2022

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

A330260 a(n) = n! * Sum_{k=0..n} binomial(n,k) * n^(n - k) / k!.

Original entry on oeis.org

1, 2, 17, 352, 13505, 830126, 74717857, 9263893892, 1513712421377, 315230799073690, 81499084718806001, 25612081645835777192, 9615370149488574778177, 4250194195208050117007942, 2184834047906975645398282625, 1292386053018890618812398220876

Offset: 0

Ilya Gutkovskiy, Dec 18 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A002720, A025167, A061711, A102757, A102773, A187021, A277373, A277452, A293146, A330497.

Main diagonal of A341014.

[Factorial(n)*&+[Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Dec 18 2019

Join[{1}, Table[n! Sum[Binomial[n, k] n^(n - k)/k!, {k, 0, n}], {n, 1, 15}]]
Join[{1}, Table[n^n n! LaguerreL[n, -1/n], {n, 1, 15}]]
Table[n! SeriesCoefficient[Exp[x/(1 - n x)]/(1 - n x), {x, 0, n}], {n, 0, 15}]

a(n) = n! * sum(k=0, n, binomial(n,k) * n^(n-k)/k!); \\ Michel Marcus, Dec 18 2019

a(n) = n! * [x^n] exp(x/(1 - n*x)) / (1 - n*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k * k!.
a(n) ~ sqrt(2*Pi) * BesselI(0,2) * n^(2*n + 1/2) / exp(n). - Vaclav Kotesovec, Dec 18 2019

A131182 Table T(n,k) = n!*k^n, read by upwards antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 8, 3, 1, 0, 24, 48, 18, 4, 1, 0, 120, 384, 162, 32, 5, 1, 0, 720, 3840, 1944, 384, 50, 6, 1, 0, 5040, 46080, 29160, 6144, 750, 72, 7, 1, 0, 40320, 645120, 524880, 122880, 15000, 1296, 98, 8, 1, 0, 362880, 10321920, 11022480, 2949120, 375000, 31104, 2058, 128, 9, 1

Offset: 0

Philippe Deléham, Sep 25 2007

For k>0, T(n,k) is the n-th moment of the exponential distribution with mean = k. - Geoffrey Critzer, Jan 06 2019

T(n,k) is the minimum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}. For the maximum value, see A331988. - Chai Wah Wu, Sep 01 2022

			The (inverted) table begins:
k=0: 1, 0,   0,    0,      0,       0, ... (A000007)
k=1: 1, 1,   2,    6,     24,     120, ... (A000142)
k=2: 1, 2,   8,   48,    384,    3840, ... (A000165)
k=3: 1, 3,  18,  162,   1944,   29160, ... (A032031)
k=4: 1, 4,  32,  384,   6144,  122880, ... (A047053)
k=5: 1, 5,  50,  750,  15000,  375000, ... (A052562)
k=6: 1, 6,  72, 1296,  31104,  933120, ... (A047058)
k=7: 1, 7,  98, 2058,  57624, 2016840, ... (A051188)
k=8: 1, 8, 128, 3072,  98304, 3932160, ... (A051189)
k=9: 1, 9, 162, 4374, 157464, 7085880, ... (A051232)
Main diagonal is 1, 1, 8, 162, 6144, 375000, ... (A061711).

Chai Wah Wu, Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized, arXiv:1508.02934 [math.CO], 2015-2020.
Chai Wah Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

Columns k=0-9 give: A000007, A000142, A000165, A032031, A047053, A052562, A047058, A051188, A051189, A051232.
Main diagonal gives A061711.
Cf. A292783, A303489, A331988.

T:= (n,k)-> n!*k^n:
seq(seq(T(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jan 06 2019

from math import factorial
def A131182_T(n, k): # compute T(n, k)
    return factorial(n)*k**n # Chai Wah Wu, Sep 01 2022

From Ilya Gutkovskiy, Aug 11 2017: (Start)
G.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - 2*k*x/(1 - 2*k*x/(1 - 3*k*x/(1 - 3*k*x/(1 - ...))))))), a continued fraction.
E.g.f. of column k: 1/(1 - k*x). (End)

A137268 Triangle T(n, k) = k! * (k+1)^(n-k), read by rows.

Original entry on oeis.org

1, 2, 2, 4, 6, 6, 8, 18, 24, 24, 16, 54, 96, 120, 120, 32, 162, 384, 600, 720, 720, 64, 486, 1536, 3000, 4320, 5040, 5040, 128, 1458, 6144, 15000, 25920, 35280, 40320, 40320, 256, 4374, 24576, 75000, 155520, 246960, 322560, 362880, 362880

Offset: 1

Roger L. Bagula, Mar 12 2008

Essentially the same as A104001.

			Triangle begins as:
    1;
    2,     2;
    4,     6,     6;
    8,    18,    24,     24;
   16,    54,    96,    120,    120;
   32,   162,   384,    600,    720,     720;
   64,   486,  1536,   3000,   4320,    5040,    5040;
  128,  1458,  6144,  15000,  25920,   35280,   40320,   40320;
  256,  4374, 24576,  75000, 155520,  246960,  322560,  362880,  362880;
  512, 13122, 98304, 375000, 933120, 1728720, 2580480, 3265920, 3628800, 3628800;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Fan Chung and R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194.

Cf. A061711, A066319, A074143, A104001, A152684.

[Factorial(k)*(k+1)^(n-k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 28 2022

T[n_, k_]:= k!*(k+1)^(n-k);
Table[T[n, k], {n, 12}, {k, n}]//Flatten

def A137268(n,k): return factorial(k)*(k+1)^(n-k)
flatten([[A137268(n,k) for k in range(1,n+1)] for n in range(14)]) # G. C. Greubel, Nov 28 2022

J(b, n) = (b+1)^(n-b)*b! if n > b, otherwise n! (notation of Chung and Graham).
From G. C. Greubel, Nov 28 2022: (Start)
T(n, k) = k! * (k+1)^(n-k).
T(n, n-2) = 2*A074143(n), n > 1.
T(2*n, n) = A152684(n).
T(2*n, n-1) = A061711(n).
T(2*n+1, n+1) = A066319(n). (End)

Edited by G. C. Greubel, Nov 28 2022

A231601 Number of permutations of [n] avoiding ascents from odd to even numbers.

Original entry on oeis.org

1, 1, 1, 4, 8, 54, 162, 1536, 6144, 75000, 375000, 5598720, 33592320, 592950960, 4150656720, 84557168640, 676457349120, 15620794116480, 140587147048320, 3628800000000000, 36288000000000000, 1035338990313196800, 11388728893445164800, 355902198372945100800

Offset: 0

Alois P. Heinz, Nov 11 2013

nonn

			a(0) = 1: ().
a(1) = 1: 1.
a(2) = 1: 21.
a(3) = 4: 132, 213, 231, 321.
a(4) = 8: 1324, 2413, 2431, 3241, 4132, 4213, 4231, 4321.
a(5) = 54: 13245, 13254, 13524, ..., 54213, 54231, 54321.
a(6) = 162: 132465, 132546, 132645, ..., 654213, 654231, 654321.

Alois P. Heinz, Table of n, a(n) for n = 0..250

Column k=0 of A231777.
Bisection gives: A061711 (even part).
Cf. A081123, A110138, A199660, A285672.

a:= n-> ceil(n/2)!*ceil(n/2)^floor(n/2):
seq(a(n), n=0..30);

a(n) = ceiling(n/2)! * ceiling(n/2)^floor(n/2).

a(n) = A081123(n+1) * A110138(n).

A062871 a(n) is the integer part of the geometric mean of n! and n^n.

Original entry on oeis.org

1, 1, 2, 12, 78, 612, 5795, 64425, 822470, 11856945, 190494094, 3374719083, 65351559893, 1373320643022, 31124359701926, 756718320351008, 19645797269948963, 542437979097898912, 15871685747774947592

Offset: 0

Olivier Gérard, Jun 26 2001

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A061711, A062872.

Mathematica
```
Floor[Sqrt[n!*n^n]]
```

a(n)={sqrtint(n! * n^n)} \\ Harry J. Smith, Aug 12 2009

from math import isqrt, factorial
def A062871(n): return isqrt(factorial(n)*n**n) # Chai Wah Wu, Jun 19 2024

A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 8, 100, 1664, 34336, 843776, 24046912, 779780096, 28357004800, 1143189536768, 50612287301632, 2441525866790912, 127479926768287744, 7163315850315825152, 431046122080208896000, 27655699473265974050816, 1884658377677216933085184

Offset: 1

Johannes W. Meijer, Jul 27 2016

nonn

The inverse multinomial transform [IML] transforms an input sequence b(n) into the output sequence a(n). The IML transform inverses the effect of the multinomial transform [MNL], see A274760, and is related to the logarithmic transform, see A274805 and the first formula.
To preserve the identity MNL[IML[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the inverse multinomial transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the inverse multinomial transform of a sequence. The first program is derived from a formula given by Alois P. Heinz for the logarithmic transform, see the first formula and A001187. The second program uses the e.g.f. for multivariate row polynomials, see A127671 and the examples. The third program uses information about the inverse of the inverse of the multinomial transform, see A274760.
The IML transform of A001818(n) = ((2*n-1)!!)^2 leads quite unexpectedly to A005411(n), a sequence related to certain Feynman diagrams.
Some IML transform pairs, n >= 1: A000110(n) and 1/A000142(n-1); A137341(n) and A205543(n); A001044(n) and A003319(n+1); A005442(n) and A000204(n); A005443(n) and A001350(n); A007559(n) and A000244(n-1); A186685(n+1) and A131040(n-1); A061711(n) and A141151(n); A000246(n) and A000035(n); A001861(n) and A141044(n-1)/A001710(n-1); A002866(n) and A000225(n); A000262(n) and A000027(n).

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = (1/0!) * (1*x(1))
a(2) = (1/1!) * (1*x(2) - x(1)^2)
a(3) = (1/2!) * (1*x(3) - 3*x(2)*x(1) + 2*x(1)^3)
a(4) = (1/3!) * (1*x(4) - 4*x(3)*x(1) - 3*x(2)^2 + 12*x(2)*x(1)^2 - 6*x(1)^4)
a(5) = (1/4!) * (1* x(5) - 5*x(4)*x(1) - 10*x(3)*x(2) + 20*x(3)*x(1)^2 + 30*x(2)^2*x(1) -60*x(2)*x(1)^3 + 24*x(1)^5)

Richard P. Feynman, QED, The strange theory of light and matter, 1985.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Logarithmic Transform.
Wikipedia, Feynman diagram

Cf. A274760, A274805, A127671, A001187, A001818, A000079, A005411.

nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: c:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*c(k), k=1..n-1)/n end: a := proc(n): c(n)/(n-1)! end: seq(a(n), n=1..nmax); # End first IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := log(1+add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(exp(add(t(n)*x^n/n, n=1..nmax)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): t(1):= b(1): for n from 2 to nmax+1 do t(n) := solve(d(n)-b(n), t(n)): a(n):=t(n): od: seq(a(n), n=1..nmax); # End third IML program.

nMax = 22; b[n_] := ((2*n-1)!!)^2; c[n_] := c[n] = b[n] - Sum[k*Binomial[n, k]*b[n-k]*c[k], {k, 1, n-1}]/n; a[n_] := c[n]/(n-1)!; Table[a[n], {n, 1, nMax}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)

a(n) = c(n)/(n-1)! with c(n) = b(n) - Sum_{k=1..n-1}(k*binomial(n, k)*b(n-k)*c(k)), n >= 1 and a(0) = undefined, with b(n) = A001818(n) = ((2*n-1)!!)^2.

a(n) = A000079(n-1) * A005411(n), n >= 1.

cp's OEIS Frontend

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

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A303489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] 1/(1 - k*x)^(n/k).

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A291699 a(n) = n^n*(2*n)!/(n!*(n + 1)!).

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A152684 a(n) is the number of top-down sequences (F_1, F_2, ..., F_n) whereas each F_i is a labeled forest on n nodes, containing i directed rooted trees. F_(i+1) is proper subset of F_i.

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A330260 a(n) = n! * Sum_{k=0..n} binomial(n,k) * n^(n - k) / k!.

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A131182 Table T(n,k) = n!*k^n, read by upwards antidiagonals.

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A137268 Triangle T(n, k) = k! * (k+1)^(n-k), read by rows.

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A291699 a(n) = n^n(2n)!/(n!*(n + 1)!).