A005225 -id:A005225 - cp's OEIS frontend

A038041 Number of ways to partition an n-set into subsets of equal size.

1, 2, 2, 5, 2, 27, 2, 142, 282, 1073, 2, 32034, 2, 136853, 1527528, 4661087, 2, 227932993, 2, 3689854456, 36278688162, 13749663293, 2, 14084955889019, 5194672859378, 7905858780927, 2977584150505252, 13422745388226152, 2, 1349877580746537123, 2

Offset: 1

Christian G. Bower

a(n) = 2 iff n is prime with a(p) = card{ 1|2|3|...|p-1|p, 123...p } = 2. - Bernard Schott, May 16 2019

			a(4) = card{ 1|2|3|4, 12|34, 14|23, 13|24, 1234 } = 5.
From _Gus Wiseman_, Jul 12 2019: (Start)
The a(6) = 27 set partitions:
  {{1}{2}{3}{4}{5}{6}}  {{12}{34}{56}}  {{123}{456}}  {{123456}}
                        {{12}{35}{46}}  {{124}{356}}
                        {{12}{36}{45}}  {{125}{346}}
                        {{13}{24}{56}}  {{126}{345}}
                        {{13}{25}{46}}  {{134}{256}}
                        {{13}{26}{45}}  {{135}{246}}
                        {{14}{23}{56}}  {{136}{245}}
                        {{14}{25}{36}}  {{145}{236}}
                        {{14}{26}{35}}  {{146}{235}}
                        {{15}{23}{46}}  {{156}{234}}
                        {{15}{24}{36}}
                        {{15}{26}{34}}
                        {{16}{23}{45}}
                        {{16}{24}{35}}
                        {{16}{25}{34}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..250
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A061095 (same but with labeled boxes), A005225, A236696, A055225, A262280, A262320.
Column k=1 of A208437.
Row sums of A200472 and A200473.
Cf. A000110, A007837 (different lengths), A035470 (equal sums), A275780, A317583, A320324, A322794, A326512 (equal averages), A326513.

A038041 := proc(n) local d;
add(n!/(d!*(n/d)!^d), d = numtheory[divisors](n)) end:
seq(A038041(n),n = 1..29); # Peter Luschny, Apr 16 2011

a[n_] := Block[{d = Divisors@ n}, Plus @@ (n!/(#! (n/#)!^#) & /@ d)]; Array[a, 29] (* Robert G. Wilson v, Apr 16 2011 *)
Table[Sum[n!/((n/d)!*(d!)^(n/d)), {d, Divisors[n]}], {n, 1, 31}] (* Emanuele Munarini, Jan 30 2014 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],SameQ@@Length/@#&]],{n,0,8}] (* Gus Wiseman, Jul 12 2019 *)

a(n):= lsum(n!/((n/d)!*(d!)^(n/d)),d,listify(divisors(n)));
makelist(a(n),n,1,40); /* Emanuele Munarini, Feb 03 2014 */

/* compare to A061095 */
mnom(v)=
/* Multinomial coefficient s! / prod(j=1, n, v[j]!) where
  s= sum(j=1, n, v[j]) and n is the number of elements in v[]. */
sum(j=1, #v, v[j])! / prod(j=1, #v, v[j]!)
A038041(n)={local(r=0);fordiv(n,d,r+=mnom(vector(d,j,n/d))/d!);return(r);}
vector(33,n,A038041(n)) /* Joerg Arndt, Apr 16 2011 */

import math
def a(n):
    count = 0
    for k in range(1, n + 1):
        if n % k == 0:
            count += math.factorial(n) // (math.factorial(k) ** (n // k) * math.factorial(n // k))
    return count # Paul Muljadi, Sep 25 2024

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

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

More terms from Erich Friedman

A055225 a(n) = Sum_{k divides n} (n/k)^k.

Original entry on oeis.org

1, 3, 4, 9, 6, 24, 8, 41, 37, 68, 12, 258, 14, 192, 384, 593, 18, 1557, 20, 2794, 2552, 2192, 24, 16730, 3151, 8388, 20440, 35394, 30, 116474, 32, 135457, 178512, 131396, 94968, 1111035, 38, 524688, 1596560, 2530986, 42, 7280934, 44, 8403778

Offset: 1

Leroy Quet, Jun 20 2000

nonn

a(n) is the number of (nonempty) linear partitions of the linearly ordered set [n] = {1,2,...,n} with blocks of the same size, where each block has exactly one element marked. For instance, for n = 4, we have the following 9 linear partitions (where the marked elements are denoted by *):
. (*)(*)(*)(*), (*2)(*4), (*234),
.               (*2)(3*), (1*34),
.               (1*)(*4), (12*4),
.               (1*)(3*), (123*).
- Emanuele Munarini, Feb 03 2014

			a(10) = 10^1 + 5^2 + 2^5 + 1^10 = 68 because positive divisors of 10 are 1, 2, 5, 10.

Nick Hobson, Table of n, a(n) for n = 1..1000

Cf. A005225, A038041, A236696.

Table[Total[Quotient[n, x = Divisors[n]]^x], {n, 44}] (* Jayanta Basu, Jul 08 2013 *)
Table[Sum[d^(n/d), {d, Divisors[n]}], {n, 1, 100}] (* Emanuele Munarini, Feb 03 2014 *)

a(n) := lsum(d^(n/d), d, listify(divisors(n))); makelist(a(n), n, 1, 40); /* Emanuele Munarini, Feb 03 2014 */

PARI
```
vector(44, n, sumdiv(n, d, (n/d)^d))
```

a(n) = sumdiv(n,d, d^(n/d) ); \\ Joerg Arndt, Apr 14 2013

G.f.: Sum_{n>=1} -log(1 - n*x^n)/n = Sum_{n>=0} a(n) x^n/n. - Paul D. Hanna, Aug 04 2002
G.f.: Sum_{n>0} n*x^n/(1-n*x^n). - Vladeta Jovovic, Sep 02 2002
Sum_{k=1..n} a(k) ~ 3^((n + 3 - mod(n,3))/3)/2. - Vaclav Kotesovec, Aug 07 2022

More terms from James Sellers, Jul 04 2000

Duplicate g.f. removed by Franklin T. Adams-Watters, Sep 01 2009

A057625 a(n) = n! * sum 1/k! where the sum is over all positive integers k that divide n.

Original entry on oeis.org

1, 3, 7, 37, 121, 1201, 5041, 62161, 423361, 5473441, 39916801, 818959681, 6227020801, 130784734081, 1536517382401, 32256486662401, 355687428096001, 10679532671808001, 121645100408832001, 3770998783116364801, 59616236292028416001, 1686001119824999577601

Offset: 1

Leroy Quet, Oct 09 2000

nonn

Sets of lists of equal size, cf. A000262. - Vladeta Jovovic, Nov 02 2003
From Gus Wiseman, Jan 10 2019: (Start)
Number of matrices whose entries are 1,...,n, up to column permutations. For example, inequivalent representatives of the a(4) = 37 matrices are:
One 1 X 4 matrix:
  [1234]
12 2 X 2 matrices:
  [12] [12] [13] [13] [14] [14] [23] [23] [24] [24] [34] [34]
  [34] [43] [24] [42] [23] [32] [14] [41] [13] [31] [12] [21]
and 24 4 X 1 matrices:
  [1][1][1][1][1][1][2][2][2][2][2][2][3][3][3][3][3][3][4][4][4][4][4][4]
  [2][2][3][3][4][4][1][1][3][3][4][4][1][1][2][2][4][4][1][1][2][2][3][3]
  [3][4][2][4][2][3][3][4][1][4][1][3][2][4][1][4][1][2][2][3][1][3][1][2]
  [4][3][4][2][3][2][4][3][4][1][3][1][4][2][4][1][2][1][3][2][3][1][2][1]
in total 1+12+24 = 37.
(End)

			a(4) = 4! (1 + 1/2! + 1/4!) = 24 (1 + 1/2 + 1/24) = 37.

Seiichi Manyama, Table of n, a(n) for n = 1..449

Cf. A000005, A005225, A038041, A038048, A061095, A121860, A209903 (exp), A236696, A320444.

Row sums of A066387.

a[n_] := n! DivisorSum[n, 1/#! &]; Array[a, 22] (* Jean-François Alcover, Dec 23 2015 *)

a(n)=n! * sumdiv(n, d, 1/d! );  /* Joerg Arndt, Oct 07 2012 */

E.g.f.: Sum_{n>0} (exp(x^n)-1). - Vladeta Jovovic, Dec 30 2001
E.g.f.: Sum_{k>0} x^k/k!/(1-x^k). - Vladeta Jovovic, Oct 14 2003
Equals the logarithmic derivative of A209903. - Paul D. Hanna, Jul 26 2012

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

A028417 Sum over all n! permutations of n elements of minimum lengths of cycles.

Original entry on oeis.org

1, 3, 10, 45, 236, 1505, 10914, 90601, 837304, 8610129, 96625970, 1184891081, 15665288484, 223149696601, 3394965018886, 55123430466945, 948479737691504, 17289345305870561, 332019600921360594, 6713316975465246889, 142321908843254560540, 3161718732648662557161

Offset: 1

Joe Keane (jgk(AT)jgk.org)

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..450

Cf. A028418, A046746, A006128.
Cf. A005225.
Column k=1 of A322383.

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, min(m,j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, infinity):
seq(a(n), n=1..25);  # Alois P. Heinz, May 14 2016

Drop[Apply[Plus,Table[nn=25;Range[0,nn]!CoefficientList[Series[Exp[Sum[ x^i/i,{i,n,nn}]]-1,{x,0,nn}],x],{n,1,nn}]],1] (* Geoffrey Critzer, Jan 10 2013 *)
b[n_, m_] := b[n, m] = If[n == 0, m, Sum[(j-1)! b[n-j, Min[m, j]]* Binomial[n-1, j-1], {j, n}]];
a[n_] := b[n, Infinity];
Array[a, 25] (* Jean-François Alcover, Apr 21 2020, after Alois P. Heinz *)

E.g.f.: Sum[k>0, -1+ exp(Sum(j>=k, x^j/j))]. - Vladeta Jovovic, Jul 26 2004

a(n) = Sum_{k=1..n} k * A145877(n,k). - Alois P. Heinz, Jul 28 2014

More terms from Vladeta Jovovic, Sep 19 2002

A132960 a(n) = n!Sum_{d|n} (-1)^(d+1)/(d!(n/d)^d).

Original entry on oeis.org

1, 0, 3, 2, 25, 94, 721, 3674, 42561, 291248, 3628801, 34254604, 479001601, 5337581534, 88966701825, 1140807642974, 20922789888001, 321094542593824, 6402373705728001, 109338195253235948, 2457732174030848001

Offset: 1

Vladeta Jovovic, Sep 06 2007

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

Cf. A005225, A132958, A132959, A132961, A132962, A132963.

Rest[ Range[0, 22]! CoefficientList[ Series[ - Sum[ Exp[ -x^k/k], {k, 25}], {x, 0, 22}], x]] (* Robert G. Wilson v, Sep 13 2007 *)

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

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

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

A236696 Number of forests on n vertices consisting of labeled rooted trees of the same size.

Original entry on oeis.org

1, 3, 10, 77, 626, 8707, 117650, 2242193, 43250842, 1049248991, 25937424602, 772559330281, 23298085122482, 817466439388341, 29223801257127976, 1181267018656911617, 48661191875666868482, 2232302772999145783735, 104127350297911241532842

Offset: 1

Emanuele Munarini, Jan 30 2014

nonn

			For n = 3 we have the following 10 forests (where the roots are denoted by ^):
                              3  2  3  1  2  1
                              |  |  |  |  |  |
         2   3  1   3  1   2  2  3  1  3  1  2
          \ /    \ /    \ /   |  |  |  |  |  |
  1 2 3    1      2      3    1  1  2  2  3  3
  ^ ^ ^,   ^,     ^,     ^,   ^, ^, ^, ^, ^, ^

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

Cf. A000169, A038041, A005225, A055225.

Table[Sum[n!/(n/d)!*(d^(d-1)/d!)^(n/d), {d,Divisors[n]}], {n,1,100}]

a(n):= lsum(n!/(n/d)!*(d^(d-1)/d!)^(n/d),d,listify(divisors(n))); makelist(a(n),n,1,40); /* Emanuele Munarini, Feb 03 2014 */

a(n) = sum(d divides n, n!/(n/d)!*(d^(d-1)/d!)^(n/d) ).

E.g.f.: sum(k>=1, exp(k^(k-1)*x^k/k!)).

A218868 Triangular array read by rows: T(n,k) is the number of n-permutations that have exactly k distinct cycle lengths.

Original entry on oeis.org

1, 2, 3, 3, 10, 14, 25, 95, 176, 424, 120, 721, 3269, 1050, 6406, 21202, 12712, 42561, 178443, 141876, 436402, 1622798, 1418400, 151200, 3628801, 17064179, 17061660, 2162160, 48073796, 177093256, 212254548, 41580000, 479001601, 2293658861, 2735287698, 719072640

Offset: 1

Geoffrey Critzer, Nov 07 2012

T(A000217(n),n) gives A246292. - Alois P. Heinz, Aug 21 2014

			:      1;
:      2;
:      3,       3;
:     10,      14;
:     25,      95;
:    176,     424,     120;
:    721,    3269,    1050;
:   6406,   21202,   12712;
:  42561,  178443,  141876;
: 436402, 1622798, 1418400, 151200;

Alois P. Heinz, Rows n = 1..170, flattened
P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009

Columns k=1-3 give: A005225, A005772, A133119.
Row sums are: A000142.
Row lengths are: A003056.
Cf. A208437, A242027 (the same for endofunctions), A246292, A317327.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(j=0, 1, x), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..16);  # Alois P. Heinz, Aug 21 2014

nn=10;a=Product[1-y+y Exp[x^i/i],{i,1,nn}];f[list_]:=Select[list,#>0&];Map[f,Drop[Range[0,nn]!CoefficientList[Series[a ,{x,0,nn}],{x,y}],1]]//Grid

E.g.f.: Product_{i>=1} (1 + y*exp(x^i/i) - y).

A346055 Expansion of e.g.f. Product_{k>=1} B(x^k/k) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 3, 10, 47, 246, 1617, 11586, 97463, 891610, 9189623, 102024396, 1250714445, 16351489116, 232261545869, 3499469551402, 56582677946675, 964734301550142, 17509882651329087, 333381717125596692, 6710286637806825557, 141167551783524139468

Offset: 0

Seiichi Manyama, Jul 02 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000110, A005225, A081066, A209903, A346056, A346057.

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k/k)-1))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, exp(x^k/k)-1))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)^d))*x^k))))

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)^d))*a(n-k)/(n-k)!));

E.g.f.: exp( Sum_{k>=1} (exp(x^k/k) - 1) ).
E.g.f.: exp( Sum_{k>=1} A005225(k)*x^k/k! ).
a(n) = (n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/d)^d)) * a(n-k)/(n-k)! for n > 0.

A241980 Number of endofunctions on [n] where all cycle lengths are equal.

Original entry on oeis.org

1, 1, 4, 24, 206, 2300, 31742, 522466, 9996478, 218088504, 5344652492, 145386399554, 4347272984936, 141737636485588, 5004538251283846, 190247639729155110, 7747479351505166738, 336492490519027631984, 15526758954835131888980, 758548951300064645742034

Offset: 0

Alois P. Heinz, Aug 10 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A005225, A061356, A212789, A242027 (column k=1).

Row sums of A243098.

with(numtheory):
b:= n-> `if`(n=0, 1, n!*add((d!*(n/d)^d)^(-1), d=divisors(n))):
a:= n-> add(binomial(n-1, j-1)*n^(n-j)*b(j), j=0..n):
seq(a(n), n=0..25);

nn=20;t[x_]:=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[1+Sum[Exp[t[x]^i/i]-1,{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Aug 11 2014 *)

a(n) = Sum_{j=0..n} C(n-1,j-1) * n^(n-j) * A005225(j).

a(n) = Sum_{k=0..n} A243098(n,k).

cp's OEIS Frontend

A038041 Number of ways to partition an n-set into subsets of equal size.

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A055225 a(n) = Sum_{k divides n} (n/k)^k.

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Mathematica

Maxima

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PARI

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A057625 a(n) = n! * sum 1/k! where the sum is over all positive integers k that divide n.

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Mathematica

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

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Mathematica

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A028417 Sum over all n! permutations of n elements of minimum lengths of cycles.

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Maple

Mathematica

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A132960 a(n) = n!*Sum_{d|n} (-1)^(d+1)/(d!*(n/d)^d).

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A236696 Number of forests on n vertices consisting of labeled rooted trees of the same size.

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A132960 a(n) = n!Sum_{d|n} (-1)^(d+1)/(d!(n/d)^d).