A036740 -id:A036740 - cp's OEIS frontend

A122221 Largest number k such that k! < (n!)^n.

2, 5, 8, 13, 19, 25, 32, 41, 50, 60, 72, 84, 97, 111, 126, 142, 159, 177, 196, 216, 237, 259, 282, 306, 330, 356, 383, 410, 439, 469, 499, 531, 563, 597, 631, 667, 703, 740, 779, 818, 858, 899, 942, 985, 1029, 1074, 1120, 1167, 1215, 1264, 1314, 1365, 1417

Offset: 2

Hugo Pfoertner, Sep 25 2006

			a(3)=5 because 5! = 120 is less than (3!)^3 = 216 whereas 6! = 720 > 216.

Chai Wah Wu, Table of n, a(n) for n = 2..10000

Cf. A036740, A121347, A122222, A382854.

a:=proc(n) local b: b:=proc(k) if k!<(n!)^n then k else fi end: max(seq(b(k),k=1..2200)) end: seq(a(n),n=2..67); # Emeric Deutsch, Oct 07 2006

s={};Do[k=1;Until[k!>=(n!)^n,k++]; AppendTo[s,k-1],{n,2,54}];s (* James C. McMahon, Oct 26 2024 *)

From Stirling's approximation, a(n) ~ n^2/2. A closer approximation for a(n) is n^2/2-c*n^2/log(n), where c = (1+log(0.5))/4 = A382854/2. - Johann Peters, Aug 23 2025

More terms from Emeric Deutsch, Oct 07 2006

A187751 a(n) = n^(n!) mod (n!)^n.

Original entry on oeis.org

0, 0, 0, 81, 225280, 7991790625, 1078848154238976, 65180706714634067542224001, 1650157594512930366268925848349310976, 66807065275536807794426016376688705273224158387201, 228020326859403543540241849077865865705999564800000000000000000000

Offset: 0

Alex Ratushnyak, Jan 03 2013

nonn

			a(3) = 3^6 mod 6^3 = 729 mod 216 = 81.

Cf. A053986, A036740.

A187751(n):=mod(n^(n!),(n!)^n)$ makelist(A187751(n),n,0,9); /* Martin Ettl, Jan 13 2013 */

import math
for n in range(12):
  f = math.factorial(n)
  print(pow(n, f, f**n))

a(n) = A053986(n) mod A036740(n).

a(10) from David Radcliffe, Jul 05 2025

A225764 Permanent of the n X n matrix M_n = [m_n(i,j)] with m_n(i,j) = Stirling2(n+i,j) for 1<=i,j<=n.

Original entry on oeis.org

1, 1, 10, 3206, 70437736, 183833539918302, 87416643970622777251260, 10762624962747767163398087106191432, 462465255409000135911575652811547463563975232544, 8991898462406411877745541835505866750273920745448784932109344640

Offset: 0

Alois P. Heinz, Jul 25 2013

nonn

			a(3) = Permanent([1, 7, 6; 1, 15, 25; 1, 31, 90]) = 3206.

Vaclav Kotesovec, Table of n, a(n) for n = 0..28 (terms 0..22 from Alois P. Heinz)
Wikipedia, Permanent (mathematics)

Cf. A036740 (determinant of M_n).

Cf. A008277, A048993.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> Stirling2(n+i, j)))):
seq(a(n), n=0..10);

a[n_] := Permanent[Table[StirlingS2[n+i, j], {i, n}, {j, n}]]; a[0] = 1; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jan 07 2016 *)

A254866 a(n) = (n!!)^n.

Original entry on oeis.org

1, 1, 4, 27, 4096, 759375, 12230590464, 140710042265625, 472769874482845188096, 601016336033894136931640625, 697127546117424200558837760000000000, 153133225508583375568553948649382367138671875, 91653624689233987245068783089656480594395136000000000000

Offset: 0

Vaclav Kotesovec, Feb 19 2015

Eric Weisstein, Double Factorial, (MathWorld)

Cf. A006882, A036740, A185141.

Cf. A000165, A001147.

Mathematica
```
Table[(n!!)^n, {n, 0, 15}]
```

a(n) ~ Pi^(n/2) * n^(n*(n+1)/2) / exp(n^2/2 - 1/6) if n is even.

a(n) ~ 2^(n/2) * n^(n*(n+1)/2) / exp(n^2/2 - 1/6) if n is odd.

A261114 Decimal expansion of Sum_{n>=0} 1/((n!)^n).

Original entry on oeis.org

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

Offset: 1

Daniel Suteu, Sep 20 2015

			2.2546326437516075174660917619423624873602862427910816970143008652092...

Table of n, a(n) for n = 1..440

Sum of 1/A036740(n).

RealDigits[N[Sum[1/((n!)^n), {n, 0, Infinity}], 120]] // First (* Michael De Vlieger, Sep 22 2015 *)

suminf(k=0, 1/k!^k) \\ Michel Marcus, Sep 20 2022

A261602 Triangular array of A(n,k) for n>=1 and 0<=k<=n^2 equal the number of permutations of the set {1,2,...,n}^2 such that first coordinates of first k elements are nondecreasing and second coordinates of the remaining n^2-k elements are nondecreasing.

Original entry on oeis.org

1, 1, 4, 8, 10, 8, 4, 216, 648, 1188, 1668, 1944, 1944, 1668, 1188, 648, 216, 331776, 1327104, 3151872, 5695488, 8608896, 11446272, 13791744, 15326208, 15858432, 15326208, 13791744, 11446272, 8608896, 5695488, 3151872, 1327104, 331776, 24883200000, 124416000000, 360806400000, 787138560000, 1426595328000, 2262299258880, 3240594432000, 4283587584000, 5304730521600, 6222411878400, 6968709089280, 7493189990400, 7763310604800

Offset: 1

Max Alekseyev, Aug 25 2015

A(n,k) = A(n,n^2-k)

It is conjectured that A(n,k)>A(n,k-1) for k<=floor(n^2/2) (see Mathoverflow link).

			The array starts with
n=1: 1, 1
n=2: 4, 8, 10, 8, 4
n=3: 216, 648, 1188, 1668, 1944, 1944, 1668, 1188, 648, 216
...

Max Alekseyev, Table of n, a(n) for n = 1..395
Mathoverflow, A question about certain sets of permutations of the ordered pairs (1,1), (1,2), ..., (n,n)

Cf. A036740 (A(n,0)), A261603 (A(n,[n^2/2])).

{ A(n,k) = my(r,rw,rs,s,t,p); r=vector(n^2+1); rw=[]; forvec(v=vector(n,i,[0,1]),rw=concat(rw,[v])); rs=vector(#rw,i,sum(j=1,n,rw[i][j])); forvec(v=vector(n,i,[1,#rw]), s=sum(j=1,#v,rs[v[j]]); t=n!; p=1; for(i=2,#v,if(v[i]==v[i-1],p++,t/=p!;p=1)); t/=p!; r[s+1]+=t*prod(i=1,n,rs[v[i]]!)*prod(j=1,n,(n-sum(i=1,n,rw[v[i]][j]))!); ,1); r[k] }

A(n,k) = SUM rs(M,1)!*...*rs(M,n)*(n-cs(M,1))!*...*(n-cs(M,n))!, where the sum is taken over n X n (0,1)-matrices with exactly k ones, rs(M,i) and cs(M,j) are the i-th row sum and the j-th column sum of M, respectively.

A351804 a(n) = [x^n] 1/Product_{j=1..n} (1 - j^n*x).

Original entry on oeis.org

1, 1, 21, 28800, 6702928485, 485036145970949475, 17284020213927891173772415260, 439885788765576174397949231373608504971360, 10926401685584312222862714944076761452123218197332439365413, 346792877099311752547903589477147000220953930332269111366383185472249165168535

Offset: 0

Alois P. Heinz, Feb 19 2022

nonn

			a(2) = (1*1)^2 + (1*2)^2 + (2*2)^2 = 1 + 4 + 16 = 21.

Alois P. Heinz, Table of n, a(n) for n = 0..26

Cf. A036740, A351800.

b:= proc(n, k, p) option remember; `if`(k=0, 1,
      add(b(j, k-1, p)*j^p, j=1..n))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..9);

Table[SeriesCoefficient[Product[1/(1 - k^n*x), {k, 1, n}], {x, 0, n}], {n, 0, 10}] (* Vaclav Kotesovec, May 13 2025 *)

a(n) = Sum_{p in {1..n}^n : p_i <= p_{i+1}} Product_{j=1..n} p_j^n.

a(n) ~ c * n^(n^2), where c = 1/QPochhammer(exp(-1)) = 1.98244090741... - Vaclav Kotesovec, May 13 2025

A366305 a(n) = Product_{k=1..n} (k^n + (k-1)^n).

Original entry on oeis.org

1, 5, 315, 555713, 47705305725, 305469864195354625, 207095306530955763265880535, 20017329298655447986400838721630926977, 357361761140807273279996172600335233468472149678425, 1481824279740988988264353294673429995981921700740921435758587890625

Offset: 1

Vaclav Kotesovec, Oct 06 2023

nonn

Cf. A036740, A323575, A323588, A323589, A366306.

Table[Product[k^n + (k-1)^n, {k, 1, n}], {n, 1, 10}]

a(n) = (n!)^n * Product_{k=1..n} (1 + (1 - 1/k)^n).
a(n) ~ n!^n * d^n, where d = exp(Integral_{x=0..1} log(1 + exp(-1/x)) dx) = 1.14183186235785012136459060138978468902610644657603999829892450823456733...
a(n) ~ (2*Pi)^(n/2) * d^n * n^(n*(2*n+1)/2) / exp(n^2 - 1/12).

A373341 Array read by ascending antidiagonals: A(n,k) is the number of acyclic de Bruijn sequences of order k and alphabet of size n, with k > 0.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 24, 216, 16, 1, 120, 331776, 10077696, 256, 1, 720, 24883200000, 12116574790945106558976, 1023490369077469249536, 65536, 1

Offset: 1

Stefano Spezia, Jun 01 2024

The 7th antidiagonal is too large to be inserted in Data.

			The array begins:
   1,      1,                       1, ...
   2,      4,                      16, ...
   6,    216,                10077696, ...
  24, 331776, 12116574790945106558976, ...
  ...

D. Condon, Yuxin Wang, and E. Yang, De Bruijn Polyominoes, arXiv:2405.18543 [math.CO], 2024. See page 5.
T. van Aardenne-Ehrenfest and N. G. de Brujin, Circuits and Trees in Oriented Linear Graphs. In: Simon Stevin 28 (1951), pp. 203-217.

Cf. A000012 (n=1), A000142 (k=1), A001146, A003992, A036740 (k=2), A373342 (antidiagonal sums), A373343 (cyclic).

A[n_,k_]:=(n!)^(n^(k-1)); Table[A[n-k+1,k],{n,6},{k,n}]//Flatten

A(n,k) = (n!)^(n^(k-1)).

A(2,n) = A001146(n-1).

A055627 From Vikram Seth's novel, "A Suitable Boy".

Original entry on oeis.org

1, 4, 216, 72576

Offset: 0

Jack Wasey (web(AT)jackwasey.com), Jun 05 2000

nonn

The sequence appears in the book, enunciated by a mathematics professor. It seems he is trying to talk about (n!)^n (A036740), but that is a different sequence.

Vikram Seth, A Suitable Boy

cp's OEIS Frontend

A122221 Largest number k such that k! < (n!)^n.

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A187751 a(n) = n^(n!) mod (n!)^n.

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Maxima

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A225764 Permanent of the n X n matrix M_n = [m_n(i,j)] with m_n(i,j) = Stirling2(n+i,j) for 1<=i,j<=n.

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A254866 a(n) = (n!!)^n.

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Mathematica

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A261114 Decimal expansion of Sum_{n>=0} 1/((n!)^n).

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PARI

A261602 Triangular array of A(n,k) for n>=1 and 0<=k<=n^2 equal the number of permutations of the set {1,2,...,n}^2 such that first coordinates of first k elements are nondecreasing and second coordinates of the remaining n^2-k elements are nondecreasing.

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PARI

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A351804 a(n) = [x^n] 1/Product_{j=1..n} (1 - j^n*x).

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Maple

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A366305 a(n) = Product_{k=1..n} (k^n + (k-1)^n).

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