A000110 -id:A000110 - cp's OEIS frontend

A134980 a(n) = Sum_{k=0..n} binomial(n,k)n^(n-k)A000110(k).

1, 2, 10, 77, 799, 10427, 163967, 3017562, 63625324, 1512354975, 40012800675, 1166271373797, 37134022033885, 1282405154139046, 47745103281852282, 1906411492286148245, 81267367663098939459, 3683790958912910588623, 176937226305157687076779

Offset: 0

Vladeta Jovovic, Feb 04 2008

Main diagonal of array "The first r-Bell numbers", p.3 of Mezo, A108087. - Jonathan Vos Post, Sep 25 2009

Number of partitions of [2n] where at least n blocks contain their own index element. a(2) = 10: 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - Alois P. Heinz, Jan 07 2022

Robert Israel, Table of n, a(n) for n = 0..385
R. Jakimczuk, Successive Derivatives and Integer Sequences, J. Int. Seq. 14 (2011) # 11.7.3.
Istvan Mezo, The r-Bell numbers, arXiv:0909.4417 [math.CO], 2009-2010.
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1.

Main diagonal of A108087.

Cf. A000110.

with(combinat): a:= n-> add(binomial(n, k)*n^(n-k)*bell(k), k=0..n):
seq(a(n), n=0..20); # Emeric Deutsch, Mar 02 2008
# Alternate:
g:= proc(n) local S;
  S:= series(exp(exp(x)+n*x-1),x,n+1);
n!*coeff(S,x,n);
end proc:
map(g, [$0..30]); # Robert Israel, Sep 29 2017
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1,
      k*b(n-1, k)+ b(n-1, k+1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021

a[n_] := n!*SeriesCoefficient[Exp[Exp[x] + n*x - 1], {x, 0, n}]; Array[a, 19, 0] (* Jean-François Alcover, Sep 28 2017, after Ilya Gutkovskiy *)
Join[{1}, Table[Sum[Binomial[n,k]*n^(n-k)*BellB[k], {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Jun 09 2020 *)

def A134980(n):
    return add(binomial(n, k)*n^(n-k)*bell_number(k) for k in (0..n))
[A134980(n) for n in (0..18)]  # Peter Luschny, May 05 2013

a(n) = exp(-1)*Sum_{k>=0} (n+k)^n/k!.
E.g.f.: A(x) = exp(-1)*Sum_{k>=0} (1+k*x)^k/k!.
a(n) = Sum_{k=0..n} Stirling1(n,k)*A000110(n+k). - Vladeta Jovovic, Nov 08 2009
a(n) = n! * [x^n] exp(exp(x) + n*x - 1). - Ilya Gutkovskiy, Sep 26 2017
a(n) ~ exp(exp(1) - 1) * n^n. - Vaclav Kotesovec, Jun 09 2020

More terms from Emeric Deutsch, Mar 02 2008

A091768 Similar to Bell numbers (A000110).

Original entry on oeis.org

1, 2, 6, 22, 92, 426, 2150, 11708, 68282, 423948, 2788230, 19341952, 141003552, 1076787624, 8589843716, 71404154928, 617151121998, 5535236798058, 51426766394244, 494145546973656, 4903432458931118, 50181840470551778, 529009041574922566

Offset: 0

Jon Perry, Mar 06 2004

nonn

Equals row sums of triangle A163946. - Gary W. Adamson, Aug 06 2009

			The Bell numbers can be generated by;
1
1 2
2 3 5
5 7 10 15
where the Bell numbers are the last entry on each line. This last entry is the first entry on the next line and then the last two entries of the previous column are added, e.g. 7=5+2, 10=7+3, 15=10+5.
This version adds ALL of the entries in the previous column to the new entry.
1
1 2
2 4 6
6 10 16 22
where 10=6+2+1+1, 16=10+2+4, 22=16+6

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Juan S. Auli and Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020.
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
Zhicong Lin, Sherry H. F. Yan, Vincular patterns in inversion sequences, Applied Mathematics and Computation (2020), Vol. 364, 124672.

Close to A074664

Cf. A000110 (Bell Numbers), A033184, A000108, A163946.

nmax=21; b = ConstantArray[0,nmax]; b[[1]]=1; Do[b[[n+1]] = Binomial[2*n, n]/(n+1) + Sum[b[[k+1]]*Binomial[2*n-k-1, n-k-1]*(k+2)/(n+1),{k,0,n-1}],{n,1,nmax-1}]; b (* Vaclav Kotesovec, Mar 13 2014 *)

v=vector(20); for (i=1,20,v[i]=vector(i)); v[1][1]=1; for (i=2,20, v[i][1]=v[i-1][i-1]; for (j=2,i, v[i][j]=v[i][j-1]+sum(k=j-1,i-1,v[k][j-1]))); for (i=1,20,print1(","v[i][i]))

a(n)=binomial(2*n,n)/(n+1)+sum(k=0,n-1,a(k)*binomial(2*n-k-1,n-k-1)*(k+2)/(n+1)) \\ Paul D. Hanna, Aug 13 2008

a(n)=local(A=1+x*O(x^n),C=serreverse(x-x^2+x^2*O(x^n))/x); for(i=0,n,A=C+x*C^2*subst(A,x,x*C));polcoeff(A,n) \\ Paul D. Hanna, Aug 13 2008

From Paul D. Hanna, Aug 13 2008: (Start)
G.f. satisfies: (1-x)*A(x-x^2) = 1 + x*A(x).
G.f. satisfies: A(x) = C(x) + x*C(x)^2*A(x*C(x)), where C(x) is the Catalan function (A000108).
a(n) = A000108(n) + Sum_{k=0..n-1} a(k)*C(2*n-k-1,n-k-1)*(k+2)/(n+1) for n>=0; eigensequence (shift left) of the Catalan triangle A033184. (End)

More terms from Vincenzo Librandi, Mar 15 2014

A224417 Least prime p such that sum_{k=0}^n B_k*x^{n-k} is irreducible modulo p, where B_k refers to the Bell number A000110(k).

Original entry on oeis.org

2, 3, 2, 11, 3, 2, 193, 113, 2, 29, 71, 167, 19, 3, 7, 13, 199, 5, 101, 59, 13, 41, 3, 359, 7, 11, 2, 31, 197, 139, 3, 59, 2, 139, 83, 37, 23, 193, 587, 199, 67, 47, 401, 41, 571, 73, 1063, 229, 1163, 47, 53, 239, 347, 223, 577, 499, 271, 269, 11, 179

Offset: 1

Zhi-Wei Sun, Apr 06 2013

nonn

Conjecture: a(n) < 4n^2-1 for all n>0.

			a(5) = 3 since the polynomial sum_{k=0}^5 B_5*x^{5-k} = x^5+x^4+2*x^3+5*x^2+15*x+52 is irreducible modulo 3 but reducible modulo 2.
Note also that a(7) = 193 < 4*7^2-1 = 195.

Zhi-Wei Sun, Table of n, a(n) for n = 1..300

Cf. A000040, A000110, A220072, A223934, A224210, A224416, A218465, A217785, A217788, A224197.

A[n_,x_]:=A[n,x]=Sum[BellB[k]*x^(n-k),{k,0,n}]
Do[Do[If[IrreduciblePolynomialQ[A[n,x],Modulus->Prime[k]]==True,Print[n," ",Prime[k]];Goto[aa]],{k,1,PrimePi[4n^2-2]}];
Print[n," ",counterexample];Label[aa];Continue,{n,1,100}]

A305846 Inverse Weigh transform of the Bell numbers (A000110).

Original entry on oeis.org

1, 2, 3, 11, 34, 138, 610, 2976, 15612, 87905, 526274, 3334988, 22270254, 156173299, 1146640394, 8791427525, 70227355786, 583283756678, 5027823752930, 44903579714037, 414877600876638, 3959945233249877, 38996757506464858, 395749369601741015, 4134132167178705732

Offset: 1

Alois P. Heinz, Jun 11 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..576

Cf. A000110, A085686, A168246, A305850, A305853.

g:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*g(n-j), j=1..n))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; g(n)-b(n, n-1) end:
seq(a(n), n=1..30);

g[n_] := g[n] = If[n == 0, 1,
     Sum[Binomial[n-1, j-1]*g[n-j], {j, 1, n}]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[Binomial[a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := a[n] = g[n] - b[n, n - 1];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 10 2022, after Alois P. Heinz *)

Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} Bell(n) * x^n.

A066562 Smallest Bell number (A000110) divisible by n, if such a number exists, otherwise 0.

Original entry on oeis.org

1, 2, 15, 52, 5, 4140, 203, 0, 4140, 4140, 10293358946226376485095653, 4140, 52, 51724158235372, 15, 0, 4506715738447323, 4140, 21147, 4140, 21147, 1052928518014714166107781298021583534928402714242132, 4140, 0, 115975, 52, 82864869804, 51724158235372, 203, 4140

Offset: 1

Amarnath Murthy, Dec 17 2001

nonn

No Bell number is divisible by 8. - John W. Layman, Jan 02 2002

Vaclav Kotesovec, Table of n, a(n) for n = 1..134
J. W. Layman, Maximum zero strings of Bell numbers modulo primes, J. Comb. Th. A40 (1985),161-168.
G. T. Williams, Numbers generated by the function e^(e^x-1), Am. Math. Monthly 52 (1945),323-327.

Cf. A000110, A327432.

b[ n_ ] := Nest[ Factor[ D[ #1, x ] ] &, Exp[ Exp[ x - 1 ] - 1 ], n ] /. (x -> 1); Do[ k = 1; While[ c = b[ k ]; !IntegerQ[ c/n ], k++ ]; Print[ c ], {n, 1, 7} ]

More terms from John W. Layman, Jan 02 2002

More terms from David Wasserman, Mar 31 2008

A168407 E.g.f.: Sum_{n>=0} (exp(2^n*x) - 1)^n/n!, an analog of the Bell numbers (A000110).

Original entry on oeis.org

1, 2, 20, 712, 91920, 44874784, 85939843136, 660213878210688, 20540390859740217600, 2592165941692975372042752, 1324271564605167892188248409088, 2730585827960928853182474922961668096

Offset: 0

Paul D. Hanna, Nov 25 2009, Nov 25 2009, Feb 16 2010

nonn

			E.g.f.: A(x) = 1 + 2*x + 20*x^2/2! + 712*x^3/3! + 91920*x^4/4! +...
A(x) = 1 + (exp(2*x) - 1) + (exp(4*x) - 1)^2/2! + (exp(8*x) - 1)^3/3! +...+ (exp(2^n*x) - 1)^n/n! +...
a(n) = coefficient of x^n/n! in Bell(x)^(2^n) where Bell(x) = exp(exp(x)-1):
Bell(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 52*x^5/5! + 203*x^6/6! +...+ A000110(n)*x^n/n! +...

Cf. A000110, A168408, A277036, A008277.

Table[BellB[n, 2^n], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2025 *)

{a(n)=local(infnty=n^4+10);round(exp(-2^n)*sum(k=0,infnty,(2^k*k)^n/k!))}

{a(n)=n!*polcoeff(sum(k=0,n,(exp(2^k*x +x*O(x^n))-1)^k/k!),n)}

{a(n)=n!*polcoeff(exp(2^n*(exp(x +x*O(x^n))-1)),n)}

{S2(n,k)=(1/k!)*sum(i=0,k,(-1)^(k-i)*binomial(k,i)*i^n)}
{a(n)=sum(k=0,n,S2(n,k)*2^(n*k))}

{a(n)=polcoeff(sum(k=0,n,(2^k*x)^k/prod(j=1,k,1-j*2^k*x+x*O(x^n))),n)}

a(n) = exp(-2^n) * Sum_{k>=0} (2^k*k)^n/k!.
a(n) = [x^n/n! ] B(x)^(2^n), where B(x) = exp(exp(x) - 1) is the e.g.f. of the Bell numbers.
a(n) = Sum_{k=0..n} S2(n,k) * 2^(n*k), where S2(n,k) = A008277(n,k) are the Stirling numbers of the second kind.
G.f.: A(x) = Sum_{n>=0} 2^(n^2) * x^n / [Product_{k=1..n} (1 - k*2^n*x)].
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jul 02 2025

A038559 a(n) = 2*A040027(n-1) + Bell(n), where Bell = A000110.

Original entry on oeis.org

1, 3, 4, 11, 33, 114, 445, 1923, 9078, 46369, 254297, 1487896, 9239135, 60615819, 418583924, 3032405831, 22979752405, 181697363626, 1495586215841, 12789423056183, 113415288869750, 1041244540823413, 9881851825756365, 96811870321650792, 977851660102425867

Offset: 0

Henry Gould

Paolo Xausa, Table of n, a(n) for n = 0..500
H. W. Gould and J. Quaintance, A linear binomial recurrence and the Bell numbers and polynomials, Applic. Anal. Discr. Math 1 (2007) 371-385.

Cf. A000110, A040027.

A038559 := proc(n)
    2*A040027(n-1)+combinat[bell](n) ;
end proc: # R. J. Mathar, Dec 20 2013
alias(PS = ListTools:-PartialSums):
A038559List := proc(len) local a, k, P, T; a := 3; P := [1]; T := [1];
for k from 1 to len-1 do
   T := [op(T), a]; P := PS([a, op(P)]); a := P[-1] od;
T end: A038559List(25); # Peter Luschny, Mar 28 2022

a[0] = 1; a[1] = 3; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1]*a[n - k], {k, n}];
Array[a, 30, 0] (* Paolo Xausa, Sep 17 2024 *)
Module[{a = 3, p = {1}}, Join[{1, a}, Table[a = Last[p = Accumulate[Prepend[p, a]]], 28]]] (* Paolo Xausa, Sep 17 2024, after Peter Luschny *)

a(0) = 1, a(1) = 3; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Jul 10 2020

A064299 a(n) = B(n)*C(n), where B(n) are Bell numbers (A000110) and C(n) are Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 4, 25, 210, 2184, 26796, 376233, 5920200, 102816714, 1947916100, 39890416020, 876478739164, 20537052247300, 510548782729680, 13407568735200525, 370553407586717490, 10742998644116921160, 325786278993936753300, 10307990595756667951830

Offset: 0

Karol A. Penson, Sep 05 2001

nonn

From Joerg Arndt, Oct 22 2012: (Start)

Number of strings of length 2*n of up to n different types t(k) of balanced parentheses, where the first appearance of type t(k) must precede the appearance of t(k+1) for all k

For example, from the 5 parenthesis string of length 3

1: ()()(); 2: ()(()); 3: (())(); 4: (()()); 5: ((())).

we obtain the B(3) * C(3) = 5 * 5 = 25 strings

1: ()()(), ()()[], ()[](), ()[][], ()[]{};

2: ()(()), ()([]), ()[()], ()[[]], ()[{}];

3: (())(), (())[], ([])(), ([])[], ([]){};

4: (()()), (()[]), ([]()), ([][]), ([]{});

5: ((())), (([])), ([()]), ([[]]), ([{}]).

(End)

Alois P. Heinz, Table of n, a(n) for n = 0..400
K. A. Penson and J.-M. Sixdeniers, Integral Representations of Catalan and Related Numbers, J. Integer Sequences, 4 (2001), #01.2.5.

Cf. A000108, A000110.

Row sums of A253180.

with(combinat):
ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
a:= n-> bell(n)*ctln(n):
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 23 2015

a[n_] := BellB[n]*CatalanNumber[n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 25 2017 *)

from itertools import count, accumulate, islice
def A064299_gen(): # generator of terms
    yield from (1,1)
    blist, b, m = (1,2), 1, 1
    for n in count(1):
        blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield b*(m := m*(4*n+2)//(n+2))
A064299_list = list(islice(A064299_gen(),20)) # Chai Wah Wu, Jun 22 2022

[bell_number(i)*catalan_number(i) for i in range(17)] # Zerinvary Lajos, Mar 14 2009

Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n) = int(x^n*sum(sqrt((4*k-x)/x)*Heaviside(4*k-x)/(k*k!), k = 1..infinity)/(2*Pi*exp(1)), x = 0..infinity); this representation is unique.

A099977 Bisection of Bell numbers, A000110.

Original entry on oeis.org

1, 5, 52, 877, 21147, 678570, 27644437, 1382958545, 82864869804, 5832742205057, 474869816156751, 44152005855084346, 4638590332229999353, 545717047936059989389, 71339801938860275191172

Offset: 0

N. J. A. Sloane, Nov 19 2004

Cf. A000110, A020557.

G:=series(exp(exp(x)-1),x=0,50): seq((2*n-1)!*coeff(G,x^(2*n-1)),n=1..18);

from itertools import accumulate, islice
def A099977_gen(): # generator of terms
    yield 1
    blist, b = (1,2), 1
    while True:
        for _ in range(2):
            blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield b
A099977_list = list(islice(A099977_gen(),30)) # Chai Wah Wu, Jun 22 2022

E.g.f.: exp(-1)*Sum_{n>=0} n*exp(n^2*x)/n!. - Vladeta Jovovic, Aug 24 2006

a(n) = exp(-1) * Sum_{k>=0} k^(2*n+1)/k!. - Ilya Gutkovskiy, Jun 13 2019

More terms from Emeric Deutsch, Dec 07 2004

A135085 a(n) = A000110(2^n).

Original entry on oeis.org

1, 2, 15, 4140, 10480142147, 128064670049908713818925644, 172134143357358850934369963665272571125557575184049758045339873395

Offset: 0

Thomas Wieder, Nov 18 2007, Nov 19 2007

nonn

Number of set partitions of all subsets of a set, Bell(2^n).

			Let S={1,2,3,...,n} be a set of n elements and let SU be the set of all subsets of S including the empty set. The number of elements of SU is |SU| = 2^n. Now form all possible set partitions from SU including the empty set. This gives a set W and its number of elements is |W| = sum((stirling2(2^n,k)), k=0..2^n) = Bell(2^n).
For S={1,2} we have SU = { {}, {1}, {2}, {1,2} } and W =
{
{{{}}, {1}, {2}, {1, 2}},
{{2}, {1, 2}, {{}, {1}}},
{{1}, {1, 2}, {{}, {2}}},
{{1}, {2}, {{}, {1, 2}}},
{{{}}, {1, 2}, {{1}, {2}}},
{{{1}, {2}}, {{}, {1, 2}}},
{{1, 2}, {{}, {1}, {2}}},
{{{}}, {2}, {{1}, {1, 2}}},
{{{1}, {1, 2}}, {{}, {2}}},
{{2}, {{}, {1}, {1, 2}}},
{{{}}, {1}, {{2}, {1, 2}}},
{{{2}, {1, 2}}, {{}, {1}}},
{{1}, {{}, {2}, {1, 2}}},
{{{}}, {{1}, {2}, {1, 2}}},
{{{}, {1}, {2}, {1, 2}}}
}
and |W| = 15.

Cf. A000079, A000110, A008277, A077585, A135084.

ZahlDerMengenAusMengeDerZerlegungenEinerMenge:=proc() local n,nend,arg,k,w; nend:=5; for n from 0 to nend do arg:=2^n; w[n]:=sum((stirling2(arg,k)), k=0..arg); od; print(w[0],w[1],w[2],w[3],w[4],w[5],w[6],w[7],w[8],w[9],w[10]); end proc;

Table[BellB[2^n],{n,0,10}] (* Geoffrey Critzer, Jan 03 2014 *)

from sympy import bell
def A135085(n): return bell(2**n) # Chai Wah Wu, Jun 22 2022

a(n) = |W| = Sum_{k=0..2^n} Stirling2(2^n,k) = Bell(2^n), where Stirling2(n) is the Stirling number of the second kind and Bell(n) is the Bell number.

a(n) = exp(-1) * Sum_{k>=0} k^(2^n)/k!. - Ilya Gutkovskiy, Jun 13 2019

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cp's OEIS Frontend

A134980 a(n) = Sum_{k=0..n} binomial(n,k)*n^(n-k)*A000110(k).

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A091768 Similar to Bell numbers (A000110).

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A224417 Least prime p such that sum_{k=0}^n B_k*x^{n-k} is irreducible modulo p, where B_k refers to the Bell number A000110(k).

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A305846 Inverse Weigh transform of the Bell numbers (A000110).

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A066562 Smallest Bell number (A000110) divisible by n, if such a number exists, otherwise 0.

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A168407 E.g.f.: Sum_{n>=0} (exp(2^n*x) - 1)^n/n!, an analog of the Bell numbers (A000110).

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A038559 a(n) = 2*A040027(n-1) + Bell(n), where Bell = A000110.

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A134980 a(n) = Sum_{k=0..n} binomial(n,k)n^(n-k)A000110(k).