A121860 -id:A121860 - cp's OEIS frontend

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

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

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

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).

A355991 a(n) = n! * Sum_{k=1..n} 1/(k! * floor(n/k)!).

Original entry on oeis.org

1, 2, 5, 12, 57, 158, 1101, 5442, 28811, 212502, 2337513, 9422306, 122489967, 1654319046, 13917499277, 111631450818, 1897734663891, 23705612782022, 450406642858401, 3091477152208002, 51404897928720023, 1130752882197523686, 26007316290543044757

Offset: 1

Seiichi Manyama, Jul 22 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..464

Row sums of A355996.

Cf. A121860, A355886, A355987, A355989, A355990.

a[n_] := n! * Sum[1/(k! * Floor[n/k]!), {k, 1, n}]; Array[a, 23] (* Amiram Eldar, Jul 22 2022 *)

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

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

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

A321662 Number of non-isomorphic multiset partitions of weight n whose incidence matrix has all distinct entries.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 13, 15, 23, 33, 49, 59, 83, 101, 133, 281, 321, 477, 655, 941, 1249, 1795, 2241, 3039, 3867, 5047, 6257, 8063, 11459, 13891, 18165, 23149, 29975, 37885, 49197, 61829, 89877, 109165, 145673, 185671, 246131, 310325, 408799, 514485, 668017, 871383

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

The incidence matrix of a multiset partition has entry (i, j) equal to the multiplicity of vertex i in part j.
Also the number of positive integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with all different entries.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(3) = 3 through a(7) = 15 multiset partitions:
  {{111}}    {{1111}}    {{11111}}    {{111111}}      {{1111111}}
  {{122}}    {{1222}}    {{11222}}    {{112222}}      {{1112222}}
  {{1}{11}}  {{1}{111}}  {{12222}}    {{122222}}      {{1122222}}
                         {{1}{1111}}  {{122333}}      {{1222222}}
                         {{11}{111}}  {{1}{11111}}    {{1223333}}
                                      {{11}{1111}}    {{1}{111111}}
                                      {{1}{11222}}    {{11}{11111}}
                                      {{11}{1222}}    {{111}{1111}}
                                      {{112}{222}}    {{1}{112222}}
                                      {{122}{222}}    {{11}{12222}}
                                      {{2}{11222}}    {{112}{2222}}
                                      {{22}{1222}}    {{122}{2222}}
                                      {{1}{11}{111}}  {{2}{112222}}
                                                      {{22}{12222}}
                                                      {{1}{11}{1111}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000219, A007716, A008289, A059201, A114736, A117433, A120733, A121860, A321653, A321659, A321660, A321661.

(* b = A121860 *) b[n_] := Sum[n!/(d! (n/d)!), {d, Divisors[n]}];
(* c = A008289 *) c[n_, k_] := c[n, k] = If[n < k || k < 1, 0, If[n == 1, 1, c[n - k, k] + c[n - k, k - 1]]];
a[n_] := If[n == 0, 1, Sum[ (b[k] + b[k + 1] - 2) c[n, k], {k, 1, n}]];
a /@ Range[0, 45] (* Jean-François Alcover, Sep 14 2019 *)

\\ here b(n) is A121860(n).
b(n)={sumdiv(n, d, n!/(d!*(n/d)!))}
seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, if(n==1, 1, b(n-1)+b(n)-2))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

a(n) = Sum_{k>=1} (A121860(k) + A121860(k+1) - 2)*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

A258899 E.g.f.: 2 - exp(2) + Sum_{n>=1} 2^n * exp(x^n) / n!.

Original entry on oeis.org

1, 2, 6, 10, 42, 34, 786, 130, 17058, 81154, 545346, 2050, 102457218, 8194, 1141636866, 72648608770, 648648065538, 131074, 111258180895746, 524290, 40892974286411778, 229774078552113154, 28890711351291906, 8388610, 3552178288049960329218, 34469355651846669074434

Offset: 0

Paul D. Hanna, Jun 20 2015

nonn

Conjecture: the sequence a(n) taken modulo a positive integer k is eventually periodic with the period dividing phi(k). For example, the sequence taken modulo 11 is [1, 2, 6, 10, 9, 1, 5, 9, 8, 7, 10, 4, 6, 10, 7, 1, 0, 9, 5, 8, 3, 4, 6, 10, 7, 1, 0, 9, 5, 8, 3, 4, 6, 10, 7, 1, 0, 9, 5, 8, 3, ...] with an apparent period of 10 (= phi(11)) starting at n = 11. - Peter Bala, Aug 03 2025

			E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 10*x^3/3! + 42*x^4/4! + 34*x^5/5! + 786*x^6/6! +...
where
A(x) = 2 - exp(2) + 2*exp(x) + 2^2*exp(x^2)/2! + 2^3*exp(x^3)/3! + 2^4*exp(x^4)/4! + 2^5*exp(x^5)/5! +...
A(x) = 2 - exp(1) + exp(2*x) + exp(2*x^2)/2! + exp(2*x^3)/3! + exp(2*x^4)/4! + exp(2*x^5)/5! +...

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A121860, A258903.

with(numtheory): seq(`if`(n=0, 1, n!*add(2^d/(d!*(n/d)!), d in divisors(n))), n = 0..25); # Peter Bala, Aug 04 2025

{a(n) = local(A=1); A = 2-exp(2) + sum(m=1,n,2^m/m!*exp(x^m +x*O(x^n))); if(n==0,1, n!*polcoeff(A,n))}
for(n=0,30, print1(a(n),", "))

{a(n) = local(A=1); A = 2-exp(1) + sum(m=1,n,1/m!*exp(2*x^m +x*O(x^n))); if(n==0,1, n!*polcoeff(A,n))}
for(n=0,30, print1(a(n),", "))

E.g.f.: 2 - exp(1) + Sum_{n>=1} exp(2*x^n) / n!.

For n >= 1, a(n) = Sum_{d divides n} 2^d * n!/(d!*(n/d)!). - Peter Bala, Aug 04 2025

A346039 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k))^(1/k!).

Original entry on oeis.org

1, -1, -1, 3, 1, 17, -119, 165, 1191, -21989, 169527, -317837, -7182779, 54452161, 292654649, -4320853051, -46883217705, 728176373539, 9943868087879, -166076498591597, -2748733072385043, 65290726021558089, 151614363753006601, -11661992771499644571

Offset: 0

Seiichi Manyama, Jul 02 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..464

Cf. A000587, A121860, A330199, A345762, A346037, A346058.

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

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

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

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

E.g.f.: exp( Sum_{k>=1} (1 - exp(x^k))/k! ).
E.g.f.: exp( -Sum_{k>=1} A121860(k)*x^k/k! ).
a(n) = -(n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/d)!)) * a(n-k)/(n-k)! for n > 0.

A326374 Irregular triangle read by rows where T(n,k) is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices, where d = A027750(n,k).

Original entry on oeis.org

1, 3, 1, 16, 1, 125, 15, 1, 1296, 1, 16807, 735, 140, 1, 262144, 1, 4782969, 76545, 1890, 1, 100000000, 112000, 1, 2357947691, 13835745, 33264, 1, 61917364224, 1, 1792160394037, 3859590735, 270670400, 35135100, 720720, 1, 56693912375296, 1, 1946195068359375

Offset: 1

Gus Wiseman, Jul 03 2019

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

			Triangle begins:
           1
           3          1
          16          1
         125         15          1
        1296          1
       16807        735        140          1
      262144          1
     4782969      76545       1890          1
   100000000     112000          1
  2357947691   13835745      33264          1
The T(4,2) = 15 hypertrees:
  {{1,4,5},{2,3,5}}
  {{1,4,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,4},{2,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,2,5},{3,4,5}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{1,3,4}}
  {{1,2,4},{3,4,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{1,4,5}}

Alois P. Heinz, Rows n = 1..185, flattened

Row sums are A320444.
Column d = 1 is A000272.
Cf. A030019, A027750, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.

T:= n-> seq(n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1), d=numtheory[divisors](n)):
seq(T(n), n=1..20);  # Alois P. Heinz, Aug 21 2019

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

T(n, k) = n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1), where d = A027750(n, k).

Edited by Peter Munn, Mar 05 2025

A346037 Expansion of e.g.f. Product_{k>=1} B(x^k)^(1/k!) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 3, 9, 41, 183, 1145, 6835, 52043, 398441, 3577291, 32395905, 342875813, 3603992759, 42817702673, 518311440987, 6897155535843, 93092680608025, 1376879589495555, 20561329595474713, 333009853668160757, 5480574201430489831, 96322698607644959065

Offset: 0

Seiichi Manyama, Jul 02 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..464

Cf. A000110, A121860, A209902, A209903, A346039, A346056.

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

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

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

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

E.g.f.: exp( Sum_{k>=1} (exp(x^k) - 1)/k! ).
E.g.f.: exp( Sum_{k>=1} A121860(k)*x^k/k! ).
a(n) = (n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/d)!)) * a(n-k)/(n-k)! for n > 0.

A352058 a(n) = Sum_{p|n, p prime} n!/(p!*(n/p)!).

Original entry on oeis.org

0, 1, 1, 6, 1, 120, 1, 840, 10080, 30240, 1, 3659040, 1, 17297280, 3632428800, 259459200, 1, 1490852563200, 1, 845092863014400, 3379030566912000, 28158588057600, 1, 2565331847811532800, 1077167364120207360000, 64764752532480000, 5001134190558105600000

Offset: 1

Seiichi Manyama, Mar 02 2022

nonn

Cf. A121860, A352012, A352059, A352060.

a[1] = 0; a[n_] := Plus @@ (n!/((p=FactorInteger[n][[;;,1]])!*(n/p)!)); Array[a, 30] (* Amiram Eldar, Mar 02 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, n!/(f[k, 1]!*(n/f[k, 1])!));

my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=1, N, isprime(k)*(exp(x^k)-1)/k!))))

E.g.f.: Sum_{p prime} (exp(x^p) - 1)/p!.

a(n) = 1 if and only if n is prime.

cp's OEIS Frontend

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

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

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A320444 Number of uniform hypertrees spanning n vertices.

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A355991 a(n) = n! * Sum_{k=1..n} 1/(k! * floor(n/k)!).

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A321662 Number of non-isomorphic multiset partitions of weight n whose incidence matrix has all distinct entries.

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Extensions

A258899 E.g.f.: 2 - exp(2) + Sum_{n>=1} 2^n * exp(x^n) / n!.

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Maple

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PARI

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A346039 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k))^(1/k!).

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