A057625 -id:A057625 - cp's OEIS frontend

A038048 a(n) = (n-1)! * sigma(n).

1, 3, 8, 42, 144, 1440, 5760, 75600, 524160, 6531840, 43545600, 1117670400, 6706022400, 149448499200, 2092278988800, 40537905408000, 376610217984000, 13871809695744000, 128047474114560000, 5109094217170944000

Offset: 1

Christian G. Bower

sigma(n) = A000203(n) is the sum of the divisors of n.
Number of labeled regular octopi (or octopuses, cycles of ordered sets all the same size).
Left edge of triangle in A008298.

			a(6) = 5! * (1 + 2 + 3 + 6) = 1440 = 6! * (1 + 1/2 + 1/3 + 1/6).

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 56 (1.4.67).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159, #10, A(n,1).

T. D. Noe, Table of n, a(n) for n = 1..100
Xiaojun Liu, Motohico Mulase, Adam Sorkin, Quantum curves for simple Hurwitz numbers of an arbitrary base curve, arXiv:1304.0015 [math.AG], 2013.
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.

Cf. A000203, A057625, A058892, A110373, A110374.

a := n -> n!*add(1/j, j=numtheory:-divisors(n)): seq(a(n), n=1..23); # Emeric Deutsch, Jul 24 2005

a[n_] := (n-1)!*DivisorSigma[1, n]; Table[a[n], {n, 20}] (* Jean-François Alcover, Mar 23 2011 *)

a(n)=(n-1)!*sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

A038048 = lambda n: factorial(n-1)*sigma(n,1)
[A038048(n) for n in (1..20)] # Peter Luschny, Jan 19 2016

a(n) = Sum_{d|n} n!/d. - Amarnath Murthy, Jul 24 2005
a(p) = (p+1)*(p-1)! if p is a prime. - Amarnath Murthy, Jul 24 2005
E.g.f.: log(f(x)), where f(x) = o.g.f. for partitions (A000041), Product_{k>=1} 1/(1 - x^k). - N. J. A. Sloane
E.g.f.: Sum_{k>0} x^k/(k*(1-x^k)). - Vladeta Jovovic, Mar 27 2005
a(n) = A000142(n-1)*A000203(n). - Omar E. Pol, Feb 26 2014

More terms from Emeric Deutsch, Jul 24 2005

Edited by N. J. A. Sloane, May 12 2008 at the suggestion of Joerg Arndt

A121860 a(n) = Sum_{d|n} n!/(d!*(n/d)!).

Original entry on oeis.org

1, 2, 2, 8, 2, 122, 2, 1682, 10082, 30242, 2, 7318082, 2, 17297282, 3632428802, 36843206402, 2, 2981705126402, 2, 1690185726028802, 3379030566912002, 28158588057602, 2, 76941821303636889602, 1077167364120207360002

Offset: 1

Vladeta Jovovic, Sep 09 2006

a(n) = 2 iff n is prime.
a(468) has 1007 decimal digits. - Michael De Vlieger, Sep 12 2018
From Gus Wiseman, Jan 10 2019: (Start)
Number of matrices whose entries are 1,...,n, up to row and column permutations. For example, inequivalent representatives of the a(4) = 8 matrices are:
  [1 2 3 4]
.
  [1 2] [1 2] [1 3] [1 3] [1 4] [1 4]
  [3 4] [4 3] [2 4] [4 2] [2 3] [3 2]
.
  [1]
  [2]
  [3]
  [4]
(End)
Conjecture: the sequence a(n) taken modulo a positive integer k >= 3 eventually becomes constant equal to 2. For example, the sequence taken modulo 11 is [1, 2, 2, 8, 2, 1, 2, 10, 6, 3, 2, 2, 2, 2, 2, 2, ...]. - Peter Bala, Aug 08 2025

Michael De Vlieger, Table of n, a(n) for n = 1..467
Jimmy Devillet, Gergely Kiss, Characterizations of biselective operations, arXiv:1806.02073 [math.RA], 2018.

Cf. A038041, A057625, A061095, A100195, A236696, A258899, A258903, A320444, A386877.

with(numtheory): seq(n!*add(1/(d!*(n/d)!), d in divisors(n)), n = 1..25); # Peter Bala, Aug 04 2025

f[n_] := Block[{d = Divisors@n}, Plus @@ (n!/(d! (n/d)!))]; Array[f, 25] (* Robert G. Wilson v, Sep 11 2006 *)
Table[DivisorSum[n, n!/(#!*(n/#)!) &], {n, 25}] (* Michael De Vlieger, Sep 12 2018 *)

a(n) = sumdiv(n, d, n!/(d!*(n/d)!)); \\ Michel Marcus, Sep 13 2018

E.g.f.: Sum_{k>0} (exp(x^k)-1)/k!.

More terms from Robert G. Wilson v, Sep 11 2006

A087906 a(n) = Sum_{d|n} (n-1)!/(d-1)!.

Original entry on oeis.org

1, 2, 3, 13, 25, 301, 721, 10921, 60481, 740881, 3628801, 106777441, 479001601, 12462690241, 134399865601, 2833553923201, 20922789888001, 892191453753601, 6402373705728001, 268633265290790401, 3652732042831872001, 102181898422712908801, 1124000727777607680001

Offset: 1

Vladeta Jovovic, Oct 15 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..450

Cf. A057625, A061095, A038041, A038048, A005225, A209902 (exp).

Array[n \[Function] DivisorSum[n, (n - 1)!/(# - 1)! &], 25] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)

a(n)=sumdiv(n,d,(n-1)!/(d-1)!); \\ Joerg Arndt, May 21 2013

E.g.f.: Sum_{k>0} (exp(x^k)-1)/k = -Sum_{k>0} log(1-x^k)/k!.

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

A132958 a(n) = n!*Sum_{d|n} (-1)^(d+1)/d!.

Original entry on oeis.org

1, 1, 7, 11, 121, 479, 5041, 18479, 423361, 1844639, 39916801, 298710719, 6227020801, 43606442879, 1536517382401, 9589093113599, 355687428096001, 4259374594675199, 121645100408832001, 1135353600039859199

Offset: 1

Vladeta Jovovic, Sep 06 2007

G. C. Greubel, Table of n, a(n) for n = 1..445

Cf. A057625, A132959, A132960, A132961, A132962, A132963.

f[n_] := Block[{d = Divisors@n}, Plus @@ (n!*(-1)^(d + 1)/d!)]; Array[f, 19] (* or *) (* Robert G. Wilson v, Sep 13 2007 *)
Rest[ Range[0, 20]! CoefficientList[ Series[ Sum[(-x)^k/(k!*(x^k - 1)), {k, 25}], {x, 0, 20}], x]] (* or *) (* Robert G. Wilson v, Sep 13 2007 *)
Rest[ Range[0, 20]! CoefficientList[ Series[ Sum[1 - Exp[ -x^k], {k, 25}], {x, 0, 20}], x]] (* Robert G. Wilson v, Sep 13 2007 *)

a(n) = n!*sumdiv(n, d, (-1)^(d+1)/d!); \\ Michel Marcus, Sep 29 2017

E.g.f.: Sum_{k>0} (-x)^k/(k!*(x^k-1)) or Sum_{k>0}(1-exp(-x^k)).

More terms from Robert G. Wilson v, Sep 13 2007

A323295 Number of ways to fill a matrix with the first n positive integers.

Original entry on oeis.org

1, 1, 4, 12, 72, 240, 2880, 10080, 161280, 1088640, 14515200, 79833600, 2874009600, 12454041600, 348713164800, 5230697472000, 104613949440000, 711374856192000, 38414242234368000, 243290200817664000, 14597412049059840000, 204363768686837760000

Offset: 0

Gus Wiseman, Jan 12 2019

nonn

			The a(4) = 72 matrices consist of:
  24 row/column permutations of [1 2 3 4]
+
  4 row/column permutations of [1 2]
                               [3 4]
+
  4 row/column permutations of [1 2]
                               [4 3]
+
  4 row/column permutations of [1 3]
                               [2 4]
+
  4 row/column permutations of [1 3]
                               [4 2]
+
  4 row/column permutations of [1 4]
                               [2 3]
+
  4 row/column permutations of [1 4]
                               [3 2]
+
  24 row/column permutations of [1]
                                [2]
                                [3]
                                [4]

Cf. A000005, A000142, A038041, A053529, A057625, A061095, A120733, A121860.

Cf. A323300, A323301, A323307, A323351.

Join[{1}, Table[DivisorSigma[0, n]*n!, {n, 30}]]

a(n) = if (n==0, 1, numdiv(n)*n!); \\ Michel Marcus, Jan 15 2019

a(n) = A000005(n) * n! for n > 0, a(0) = 1.

E.g.f.: 1 + Sum_{k>=1} x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 13 2019

A209903 E.g.f.: Product_{n>=1} B(x^n) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 4, 17, 111, 752, 6893, 64171, 733540, 8751579, 119847295, 1716294780, 27583937857, 460405876777, 8428298492136, 160944930254405, 3309210789416387, 70814345769448444, 1617322515279759301, 38322855872232745163, 960820910852189283072

Offset: 0

Paul D. Hanna, Mar 15 2012

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 17*x^3/3! + 111*x^4/4! + 752*x^5/5! +...
Let B(x) = exp(exp(x)-1) be the e.g.f. of Bell numbers:
B(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 52*x^5/5! + 203*x^6/6! +...
then the e.g.f. of this sequence equals the infinite product:
A(x) = B(x)*B(x^2)*B(x^3)*B(x^4)*B(x^5)*B(x^6)...
The logarithm of the e.g.f. A(x) begins:
log(A(x)) = x + 3*x^2/2! + 7*x^3/3! + 37*x^4/4! + 121*x^5/5! + 1201*x^6/6! +...+ A057625(n)*x^n/n! +...

Seiichi Manyama, Table of n, a(n) for n = 0..443

Cf. A057625 (log), A209902, A330199.

{a(n)=local(Bell=exp(exp(x+x*O(x^n))-1));n!*polcoeff(prod(m=1,n,subst(Bell,x,x^m+x*O(x^n))),n)}

{a(n)=n!*polcoeff(exp(sum(m=1,n,x^m/m!/(1-x^m+x*O(x^n)))),n)}
for(n=0,25,print1(a(n),", "))

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/d!)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Jul 02 2021

E.g.f.: exp( Sum_{n>=1} x^n/n! / (1-x^n) ).
E.g.f.: exp( Sum_{n>=1} A057625(n)*x^n/n! ).
E.g.f.: exp( Sum_{n>=1} exp(x^n)-1 ).
a(n) = (n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/d!) * a(n-k)/(n-k)! for n > 0. - Seiichi Manyama, Jul 02 2021

A354843 a(n) = n! * Sum_{d|n} (n/d)^d / d!.

Original entry on oeis.org

1, 5, 19, 145, 601, 8521, 35281, 672001, 4898881, 82615681, 439084801, 21138606721, 80951270401, 3358578263041, 49506372115201, 1227603183206401, 6046686277632001, 611515751899852801, 2311256907767808001, 254421414038266675201, 4015778465971464192001

Offset: 1

Seiichi Manyama, Jun 08 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..447

Cf. A057625, A327579, A354845.

a[n_] := n! * DivisorSum[n, (n/#)^#/#! &]; Array[a, 20] (* Amiram Eldar, Jun 08 2022 *)

PARI
```
a(n) = n!*sumdiv(n, d, (n/d)^d/d!);
```

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, exp(k*x^k)-1)))

E.g.f.: Sum_{k>0} (exp(k * x^k) - 1).

If p is prime, a(p) = 1 + p * p!.

A327578 a(n) = n! * Sum_{d|n} d^(n/d - 1) / d!.

Original entry on oeis.org

1, 3, 7, 49, 121, 2521, 5041, 208321, 907201, 32810401, 39916801, 10621860481, 6227020801, 2877004690561, 19233710496001, 1415779600435201, 355687428096001, 1085522620595212801, 121645100408832001, 653741050484890368001, 6259137133527742464001, 576612373659657208473601

Offset: 1

Ilya Gutkovskiy, Sep 17 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..425

Cf. A057625, A087909, A327579.

a[n_] := n! Sum[d^(n/d - 1)/d!, {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[x^k/(k! (1 - k x^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n) = n! * sumdiv(n, d, d^(n/d - 1) / d!); \\ Michel Marcus, Sep 17 2019

E.g.f.: Sum_{k>=1} x^k / (k! * (1 - k * x^k)).

A320444 Number of uniform hypertrees spanning n vertices.

Original entry on oeis.org

1, 1, 1, 4, 17, 141, 1297, 17683, 262145, 4861405, 100112001, 2371816701, 61917364225, 1796326510993, 56693912375297, 1947734359001551, 72059082110369793, 2863257607266475419, 121439531096594251777, 5480987217944109919765, 262144000000000000000001

Offset: 0

Gus Wiseman, Jan 09 2019

nonn

The density of a hypergraph is the sum of sizes of its edges minus the number of edges minus the number of vertices. A hypertree is a connected hypergraph of density -1. A hypergraph is uniform if its edges all have the same size. The span of a hypergraph is the union of its edges.

			Non-isomorphic representatives of the 5 unlabeled uniform hypertrees on 5 vertices and their multiplicities in the labeled case, which add up to a(5) = 141:
   5 X {{1,5},{2,5},{3,5},{4,5}}
  60 X {{1,4},{2,5},{3,5},{4,5}}
  60 X {{1,3},{2,4},{3,5},{4,5}}
  15 X {{1,2,5},{3,4,5}}
   1 X {{1,2,3,4,5}}

Robert Israel, Table of n, a(n) for n = 0..387

Row sums of A326374.

Cf. A000272, A030019, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.

f:= proc(n) local d; add((n-1)!/(d! * ((n-1)/d)!) * (n/d)^((n-1)/d - 1), d = numtheory:-divisors(n-1)); end proc:
f(0):= 1: f(1):= 1:
map(f, [$0..25]); # Robert Israel, Jan 10 2019

Table[Sum[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1),{d,Divisors[n]}],{n,10}]

a(n) = if (n<2, 1, n--; sumdiv(n, d, n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1))); \\ Michel Marcus, Jan 10 2019

a(n + 1) = Sum_{d|n} n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1).

a(p prime) = 1 + (p + 1)^(p - 1).

A355886 a(n) = n! * Sum_{k=1..n} floor(n/k)/k!.

Original entry on oeis.org

1, 5, 22, 125, 746, 5677, 44780, 420401, 4206970, 47543141, 562891352, 7573655905, 104684547566, 1596368400005, 25482043382476, 439969180782017, 7835163501390290, 151712475696833221, 3004182138648663200, 63854641556089628801, 1400563708969910620822

Offset: 1

Seiichi Manyama, Jul 20 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..448
Vaclav Kotesovec, Plot of a(n)/(n!*n) for n = 1..10000

Cf. A006218, A057625, A356010, A356458, A229837.

Cf. A356009, A356459.

Table[n! * Sum[Floor[n/k]/k!, {k,1,n}], {n,1,25}] (* Vaclav Kotesovec, Aug 11 2025 *)

PARI
```
a(n) = n!*sum(k=1, n, n\k/k!);
```

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-x^k)))/(1-x)))

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, exp(x^k)-1)/(1-x)))

a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/d!)); \\ Seiichi Manyama, Aug 08 2022

E.g.f.: (1/(1-x)) * Sum_{k>0} x^k/(k! * (1 - x^k)).
E.g.f.: (1/(1-x)) * Sum_{k>0} (exp(x^k) - 1).
a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/d! = n! * Sum_{k=1..n} A057625(k)/k!. - Seiichi Manyama, Aug 08 2022
a(n) ~ A229837 * n! * n. - Vaclav Kotesovec, Aug 11 2025

cp's OEIS Frontend

A038048 a(n) = (n-1)! * sigma(n).

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A121860 a(n) = Sum_{d|n} n!/(d!*(n/d)!).

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A087906 a(n) = Sum_{d|n} (n-1)!/(d-1)!.

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A132958 a(n) = n!*Sum_{d|n} (-1)^(d+1)/d!.

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A323295 Number of ways to fill a matrix with the first n positive integers.

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A209903 E.g.f.: Product_{n>=1} B(x^n) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

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A354843 a(n) = n! * Sum_{d|n} (n/d)^d / d!.

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