A038041 -id:A038041 - cp's OEIS frontend

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

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

A336127 Number of ways to split a composition of n into contiguous subsequences with different sums.

Original entry on oeis.org

1, 1, 2, 8, 16, 48, 144, 352, 896, 2432, 7168, 16896, 46080, 114688, 303104, 843776, 2080768, 5308416, 13762560, 34865152, 87818240, 241172480, 583008256, 1503657984, 3762290688, 9604956160, 23689428992, 60532195328, 156397207552, 385137770496, 967978254336

Offset: 0

Gus Wiseman, Jul 09 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(4) = 16 splits:
  ()  (1)  (2)    (3)        (4)
           (1,1)  (1,2)      (1,3)
                  (2,1)      (2,2)
                  (1,1,1)    (3,1)
                  (1),(2)    (1,1,2)
                  (2),(1)    (1,2,1)
                  (1),(1,1)  (1),(3)
                  (1,1),(1)  (2,1,1)
                             (3),(1)
                             (1,1,1,1)
                             (1),(1,2)
                             (1),(2,1)
                             (1,2),(1)
                             (2,1),(1)
                             (1),(1,1,1)
                             (1,1,1),(1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal instead of different sums is A074854.
Starting with a strict composition gives A336128.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336130, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}]

a(n) = Sum_{k=0..n} 2^(n-k) k! A008289(n,k).

A326515 Number of factorizations of n into factors > 1 where every factor has the same average of prime indices.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 7, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(900) = 9 factorizations:
  (3*3*10*10),
  (3*3*100), (3*10*30), (9*10*10),
  (3*300), (9*100), (10*90), (30*30),
  (900).

Cf. A001055, A038041, A051293, A321455, A321469, A322794, A326512, A326514, A326516, A326520, A326536.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],SameQ@@Mean/@primeMS/@#&]],{n,100}]

avgpis(n) = { my(f=factor(n)); f[,1] = apply(primepi,f[,1]); (1/bigomega(n))*sum(i=1,#f~,f[i,2]*f[i,1]); };
has_same_average_of_pis(facs) = if(!#facs, 1, my(avg=0); for(i=1,#facs,if(!avg, avg=avgpis(facs[i]), if(avg!=avgpis(facs[i]), return(0)))); (1));
A326515(n, m=n, facs=List([])) = if(1==n, has_same_average_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A326515(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A326517 Number of normal multiset partitions of weight n where each part has a different size.

Original entry on oeis.org

1, 1, 2, 12, 28, 140, 956, 3520, 17792, 111600, 1144400, 4884064, 34907936, 214869920, 1881044032, 25687617152, 139175009920, 1098825972608, 8770328141888, 74286112885504, 784394159958848, 15114871659653952, 92392468773724544, 889380453354852416, 7652770202041529856

Offset: 0

Gus Wiseman, Jul 12 2019

nonn

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(3) = 12 normal multiset partitions:
  {}  {{1}}  {{1,1}}  {{1,1,1}}
             {{1,2}}  {{1,1,2}}
                      {{1,2,2}}
                      {{1,2,3}}
                      {{1},{1,1}}
                      {{1},{1,2}}
                      {{1},{2,2}}
                      {{1},{2,3}}
                      {{2},{1,1}}
                      {{2},{1,2}}
                      {{2},{1,3}}
                      {{3},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Row sums of A332253.

Cf. A007837, A038041, A255906, A317583, A326026, A326514, A326518, A326519, A326520, A326521, A326533.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..min(1, n/i))))
    end:
a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..n), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 23 2023

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Length/@#&]],{n,0,6}]

R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}
seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ Andrew Howroyd, Feb 07 2020

Terms a(8) and beyond from Andrew Howroyd, Feb 07 2020

A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 29, 37, 57, 89, 265, 309, 521, 745, 1129, 3005, 3545, 5685, 8201, 12265, 16629, 41369, 48109, 77265, 107645, 160681, 214861, 316913, 644837, 798861, 1207445, 1694269, 2437689, 3326705, 4710397, 6270513, 12246521, 14853625, 22244569, 30308033, 43706705, 57926577, 82166105, 107873221, 148081785, 257989961, 320873065, 458994657, 628016225, 875485585, 1165065733

Offset: 0

Gus Wiseman, Jul 10 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(5) = 5 splits:
  ()  (1)  (2)  (3)     (4)     (5)
                (12)    (13)    (14)
                (21)    (31)    (23)
                (1)(2)  (1)(3)  (32)
                (2)(1)  (3)(1)  (41)
                                (1)(4)
                                (2)(3)
                                (3)(2)
                                (4)(1)
The a(6) = 29 splits:
  (6)    (1)(5)   (1)(2)(3)
  (15)   (2)(4)   (1)(3)(2)
  (24)   (4)(2)   (2)(1)(3)
  (42)   (5)(1)   (2)(3)(1)
  (51)   (1)(23)  (3)(1)(2)
  (123)  (1)(32)  (3)(2)(1)
  (132)  (13)(2)
  (213)  (2)(13)
  (231)  (2)(31)
  (312)  (23)(1)
  (321)  (31)(2)
         (32)(1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal instead of different sums is A336130.
Starting with a non-strict composition gives A336127.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Set partitions with distinct block-sums are A275780.
Compositions of partitions are A323583.
Cf. A006951, A063834, A271619, A279375, A305551, A326519, A317508, A318684, A336133, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]

a(31)-a(50) from Max Alekseyev, Feb 14 2024

A061095 Number of ways of dividing n labeled items into labeled boxes with an equal number of items in each box.

Original entry on oeis.org

1, 3, 7, 31, 121, 831, 5041, 42911, 364561, 3742453, 39916801, 486891175, 6227020801, 87859375033, 1307843292757, 21004582611871, 355687428096001, 6415015584161757, 121645100408832001, 2435278206317164781, 51091124681475552961, 1124549556257968545433

Offset: 1

Henry Bottomley, May 29 2001

			a(6) = 720+90+20+1 = 831 since 720 ways of evenly distributing six labeled items into six labeled boxes, 90 into three, 20 into two and 1 into one.

Seiichi Manyama, Table of n, a(n) for n = 1..449
Kévin Perrot, Sylvain Sené, and Léah Tapin, On countings and enumerations of block-parallel automata networks, arXiv:2304.09664 [cs.DM], 2023.

Cf. A038041, A060538.

Column k=1 of A327803.

A061095 := n -> add(n!/(n/d)!^d, d = numtheory[divisors](n));
seq(A061095 (n), n = 1..17); # Peter Luschny, Apr 13 2011

Table[Sum[n!/(n/d)!^d,{d,Divisors[n]}],{n,1,20}]  (* Geoffrey Critzer, Aug 18 2011 *)

/* compare to A038041 */
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]!)
A061095(n)={local(r=0);fordiv(n,d,r+=mnom(vector(d,j,n/d)));return(r);}
vector(33,n,A061095(n)) /* Joerg Arndt, Apr 16 2011 */

a(n)=sumdiv(n,d, n!/(n/d)!^d ); \\ Joerg Arndt, Feb 23 2014

a(n) = Sum_{d|n} n!/(n/d)!^d.
E.g.f.: Sum_{k>0} x^k/(k!-x^k). - Vladeta Jovovic, Oct 13 2003
a(n) ~ n!. - Vaclav Kotesovec, Jul 21 2019

A322794 Number of factorizations of n into factors > 1 where all factors have the same number of prime factors counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 4, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 3, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 26 2018

nonn

Also the number of uniform multiset partitions of the multiset of prime indices of n, where a multiset partition is uniform if all parts have the same size.

			The a(1260) = 13 factorizations:
  (1260)  (18*70)   (4*9*35)   (2*2*3*3*5*7)
          (20*63)   (6*6*35)
          (28*45)   (4*15*21)
          (30*42)   (6*10*21)
          (12*105)  (6*14*15)
                    (9*10*14)
The a(1260) = 13 multiset partitions:
  {{1},{1},{2},{2},{3},{4}}
     {{1,1},{2,2},{3,4}}
     {{1,1},{2,3},{2,4}}
     {{1,2},{1,2},{3,4}}
     {{1,2},{1,3},{2,4}}
     {{1,2},{1,4},{2,3}}
     {{2,2},{1,3},{1,4}}
      {{1,1,2},{2,3,4}}
      {{1,2,2},{1,3,4}}
      {{1,1,3},{2,2,4}}
      {{1,1,4},{2,2,3}}
      {{1,2,3},{1,2,4}}
       {{1,1,2,2,3,4}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A001055, A001222, A038041, A112798, A306017, A306021, A319169, A320324, A317583, A321455, A321469, A326514, A326515, A326516.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],SameQ@@PrimeOmega/@#&]],{n,100}]

A299471 Regular triangle where T(n,k) is the number of labeled k-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 41, 11, 1, 1, 768, 958, 26, 1, 1, 27449, 1042642, 32596, 57, 1, 1, 1887284, 34352419335, 34359509614, 2096731, 120, 1, 1, 252522481, 72057319189324805, 1180591620442534312297, 72057594021152435, 268434467, 247, 1, 1, 66376424160

Offset: 1

Gus Wiseman, Jun 18 2018

			Triangle begins:
  1;
  1,     1;
  1,     4,       1;
  1,    41,      11,     1;
  1,   768,     958,    26,  1;
  1, 27449, 1042642, 32596, 57, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..91 (rows 1..13)
Wikipedia, Hypergraph

Columns 1..4 are A000012, A006129, A302374, A302396.
Row sums are A306021.
The unlabeled version is A301922.
The connected version is A299354.
Cf. A000005, A001315, A006126, A038041, A298422, A298426, A306017, A306018, A306019, A306020.

Table[Sum[(-1)^(n-d)*Binomial[n,d]*2^Binomial[d,k],{d,0,n}],{n,10},{k,n}]

T(n, k) = sum(d = 0, n, (-1)^(n-d)*binomial(n,d)*2^binomial(d,k)) \\ Andrew Howroyd, Jan 16 2024

T(n, k) = Sum_{d = 0..n} (-1)^(n-d)*binomial(n,d)*2^binomial(d,k).

A326512 Number of set partitions of {1..n} where every block has the same average.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 18, 16, 75, 64, 405, 302, 2581, 1693, 19872, 11295, 175807, 87524, 1851135, 787515, 21909766, 8185713, 298698113, 96514608, 4538610230, 1285072142

Offset: 0

Gus Wiseman, Jul 11 2019

The common average is necessarily (n+1)/2. The number of blocks with this average is given by A070925. - Christian Sievers, Aug 22 2024

			The a(1) = 1 through a(7) = 18 set partitions:
  {1}  {12}  {123}    {1234}    {12345}      {123456}      {1234567}
             {13}{2}  {14}{23}  {1245}{3}    {1256}{34}    {123567}{4}
                                {135}{24}    {1346}{25}    {12467}{35}
                                {15}{234}    {16}{2345}    {1267}{345}
                                {15}{24}{3}  {16}{25}{34}  {13457}{26}
                                                           {1357}{246}
                                                           {1456}{237}
                                                           {147}{2356}
                                                           {156}{2347}
                                                           {17}{23456}
                                                           {1267}{35}{4}
                                                           {1357}{26}{4}
                                                           {147}{26}{35}
                                                           {156}{237}{4}
                                                           {17}{2356}{4}
                                                           {17}{246}{35}
                                                           {17}{26}{345}
                                                           {17}{26}{35}{4}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A000110, A007837, A035470, A038041, A070925, A275780, A306017, A306021, A319169, A326513, A326515, A326520, A326536.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],SameQ@@Mean/@#&]],{n,0,8}]

a(12)-a(15) from Alois P. Heinz, Jul 12 2019

a(16)-a(26) from Christian Sievers, Aug 22 2024

A336130 Number of ways to split a strict composition of n into contiguous subsequences all having the same sum.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 15, 13, 23, 27, 73, 65, 129, 133, 241, 375, 519, 617, 1047, 1177, 1859, 2871, 3913, 4757, 7653, 8761, 13273, 16155, 28803, 30461, 50727, 55741, 87743, 100707, 152233, 168425, 308937, 315973, 500257, 571743, 871335, 958265, 1511583, 1621273, 2449259, 3095511, 4335385, 4957877, 7554717, 8407537, 12325993, 14301411, 20348691, 22896077, 33647199, 40267141, 56412983, 66090291, 93371665, 106615841, 155161833

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(7) = 13 splits:
  (1)  (2)  (3)    (4)    (5)    (6)        (7)
            (1,2)  (1,3)  (1,4)  (1,5)      (1,6)
            (2,1)  (3,1)  (2,3)  (2,4)      (2,5)
                          (3,2)  (4,2)      (3,4)
                          (4,1)  (5,1)      (4,3)
                                 (1,2,3)    (5,2)
                                 (1,3,2)    (6,1)
                                 (2,1,3)    (1,2,4)
                                 (2,3,1)    (1,4,2)
                                 (3,1,2)    (2,1,4)
                                 (3,2,1)    (2,4,1)
                                 (1,2),(3)  (4,1,2)
                                 (2,1),(3)  (4,2,1)
                                 (3),(1,2)
                                 (3),(2,1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with different instead of equal sums is A336128.
Starting with a non-strict composition gives A074854.
Starting with a partition gives A317715.
Starting with a strict partition gives A318683.
Set partitions with equal block-sums are A035470.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A271619, A279375, A305551, A317508, A318684, A326519, A336127, A336132, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],SameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]

a(31)-a(60) from Max Alekseyev, Feb 14 2024

cp's OEIS Frontend

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

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A336127 Number of ways to split a composition of n into contiguous subsequences with different sums.

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A326515 Number of factorizations of n into factors > 1 where every factor has the same average of prime indices.

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A326517 Number of normal multiset partitions of weight n where each part has a different size.

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A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.

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A061095 Number of ways of dividing n labeled items into labeled boxes with an equal number of items in each box.

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PARI

PARI

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A322794 Number of factorizations of n into factors > 1 where all factors have the same number of prime factors counted with multiplicity.

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