A182927 -id:A182927 - cp's OEIS frontend

A007837 Number of partitions of n-set with distinct block sizes.

1, 1, 1, 4, 5, 16, 82, 169, 541, 2272, 17966, 44419, 201830, 802751, 4897453, 52275409, 166257661, 840363296, 4321172134, 24358246735, 183351656650, 2762567051857, 10112898715063, 62269802986835, 343651382271526, 2352104168848091, 15649414071734847

Offset: 0

Arnold Knopfmacher

nonn

Conjecture: the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Cf. A185895. - Peter Bala, Mar 17 2022

			From _Gus Wiseman_, Jul 13 2019: (Start)
The a(1) = 1 through a(5) = 16 set partitions with distinct block sizes:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}    {{1,2,3,4,5}}
                  {{1},{2,3}}  {{1},{2,3,4}}  {{1},{2,3,4,5}}
                  {{1,2},{3}}  {{1,2,3},{4}}  {{1,2},{3,4,5}}
                  {{1,3},{2}}  {{1,2,4},{3}}  {{1,2,3},{4,5}}
                               {{1,3,4},{2}}  {{1,2,3,4},{5}}
                                              {{1,2,3,5},{4}}
                                              {{1,2,4},{3,5}}
                                              {{1,2,4,5},{3}}
                                              {{1,2,5},{3,4}}
                                              {{1,3},{2,4,5}}
                                              {{1,3,4},{2,5}}
                                              {{1,3,4,5},{2}}
                                              {{1,3,5},{2,4}}
                                              {{1,4},{2,3,5}}
                                              {{1,4,5},{2,3}}
                                              {{1,5},{2,3,4}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..700
Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, Fig. 3, arXiv:math/0606370 [math.CO], 2006.
Knopfmacher, A., Odlyzko, A. M., Pittel, B., Richmond, L. B., Stark, D., Szekeres, G. and Wormald, N. C., The asymptotic number of set partitions with unequal block sizes, Electron. J. Combin., 6 (1999), no. 1, Research Paper 2, 36 pp.
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Row sums of A131632 or A262072 or A262078 or A309992.
Cf. A000110, A005651, A007838, A032011, A035470, A038041, A178682, A265950, A271423, A275780, A326026, A326514, A326517, A326533.
Column k=0 of A327869.

a:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*a(n-k), k=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 06 2008
# second Maple program:
A007837 := proc(n) option remember; local k; `if`(n = 0, 1,
add(binomial(n-1, k-1) * A182927(k) * A007837(n-k), k = 1..n)) end:
seq(A007837(i),i=0..24); # Peter Luschny, Apr 25 2011

nn=20;p=Product[1+x^i/i!,{i,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[p,{x,0,nn}],x],1]  (* Geoffrey Critzer, Sep 22 2012 *)
a[0]=1; a[n_] := a[n] = Sum[(n-1)!/(n-k)!*DivisorSum[k, -#*(-#!)^(-k/#)&]* a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Vladeta Jovovic *)

{my(n=20); Vec(serlaplace(prod(k=1, n, (1+x^k/k!) + O(x*x^n))))} \\ Andrew Howroyd, Dec 21 2017

E.g.f.: Product_{m >= 1} (1+x^m/m!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} (-d)*(-d!)^(-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 13 2002
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

More terms from Christian G. Bower

a(0)=1 prepended by Alois P. Heinz, Aug 29 2015

A182926 Row sums of absolute values of A182928.

Original entry on oeis.org

1, 2, 3, 10, 25, 161, 721, 5706, 40881, 385687, 3628801, 41268613, 479001601, 6324319717, 87212177053, 1317906346186, 20922789888001, 357099708702023, 6402373705728001, 121882752536893635, 2432928081076384321, 51140835669924352717

Offset: 1

Peter Luschny, Apr 16 2011

nonn

The sum of multinomial coefficients can be computed recursively as
A005651(0) = 1 and A005651(n) = Sum_{1<=k<=n} binomial(n-1,k-1) * A182926(k) * A005651(n-k).
Möbius inversion yields: 1, 1, 2, 8, 24, 157, 720, 5696, 40878,...
A182927(2*i+1) = A182926(2*i+1).

			a(6) = 1 + 10 + 30 + 120 = 161.

Seiichi Manyama, Table of n, a(n) for n = 1..450

Cf. A182927, A182928, A005651, A007837.

A182926 := proc(n) local d;
add(n!/(d*((n/d)!)^d),d = numtheory[divisors](n)) end:
seq(A182926(i), i = 1..22);

a[n_] := Sum[ Abs[ -n!/(d*(-(n/d)!)^d)], {d, Divisors[n]}]; Table[ a[n], {n, 1, 22}] (* Jean-François Alcover, Jul 29 2013 *)

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

E.g.f.: Sum_{k>=1} log(1/(1 - x^k/k!)). - Ilya Gutkovskiy, May 21 2019

A308345 Expansion of e.g.f. Sum_{k>=1} log(1/(1 - x^k/k)).

Original entry on oeis.org

1, 2, 4, 15, 48, 310, 1440, 11970, 85120, 821016, 7257600, 91707000, 958003200, 13440913200, 178919989248, 2809456650000, 41845579776000, 763629026160000, 12804747411456000, 257140635922025856, 4918792391884800000, 106876408948152480000

Offset: 1

Ilya Gutkovskiy, May 21 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..450

Cf. A007841, A038048, A182926, A182927, A308337.

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

a(n) = n! * Sum_{d|n} 1/(d*(n/d)^d).
a(n) = A007841(n) - (1/n) * Sum_{k=1..n-1} k*binomial(n,k)*A007841(n-k)*a(k).
a(n) ~ 2 * (n-1)!. - Vaclav Kotesovec, Feb 16 2020

A308336 Expansion of e.g.f. exp(-1 + Product_{k>=1} (1 + x^k/k!)).

Original entry on oeis.org

1, 1, 2, 8, 31, 147, 884, 5567, 39176, 311400, 2644490, 24206327, 239684768, 2519262527, 28077597357, 331892965533, 4130002336563, 53944450834303, 738940309779760, 10577568411051305, 157846971489443335, 2452481386778640564, 39589449956634478543

Offset: 0

Ilya Gutkovskiy, May 20 2019

nonn

Cf. A007837, A143463, A182927, A308338.

nmax = 22; CoefficientList[Series[Exp[Product[(1 + x^k/k!), {k, 1, nmax}] - 1], {x, 0, nmax}], x] Range[0, nmax]!

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A007837(k)*a(n-k).

A328193 Expansion of e.g.f. Sum_{k>=1} log(1/(1 + (-x)^k/k)).

Original entry on oeis.org

1, 0, 4, 3, 48, 10, 1440, 1890, 85120, 49896, 7257600, 6883800, 958003200, 792277200, 178919989248, 194107914000, 41845579776000, 29714949264000, 12804747411456000, 12900082757417856, 4918792391884800000, 4594737608304480000, 2248001455555215360000

Offset: 1

Ilya Gutkovskiy, Oct 30 2019

nonn

Georg Fischer, Table of n, a(n) for n = 1..100

Cf. A007838, A132960, A182927, A292358, A308338, A308345.

a:= n-> n!*add((-1)^(n-d)/(d*(n/d)^d), d=numtheory[divisors](n)):
seq(a(n), n=1..24);  # Alois P. Heinz, Oct 30 2019

nmax = 23; CoefficientList[Series[Sum[Log[1/(1 + (-x)^k/k)], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! Sum[(-1)^(n - d)/(d (n/d)^d), {d, Divisors[n]}], {n, 1, 23}]

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

cp's OEIS Frontend

A007837 Number of partitions of n-set with distinct block sizes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A182926 Row sums of absolute values of A182928.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A308345 Expansion of e.g.f. Sum_{k>=1} log(1/(1 - x^k/k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A308336 Expansion of e.g.f. exp(-1 + Product_{k>=1} (1 + x^k/k!)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A328193 Expansion of e.g.f. Sum_{k>=1} log(1/(1 + (-x)^k/k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula