A000110 -id:A000110 - cp's OEIS frontend

A144293 Largest prime factor of n-th Bell number A000110(n) (or 1 if A000110(n) = 1).

1, 1, 2, 5, 5, 13, 29, 877, 23, 53, 4639, 22619, 2423, 27644437, 1800937, 1101959, 43486067, 255755771, 5006399, 222527, 4326209287, 188633, 574631, 13369534669, 1204457631577, 171659, 11759883224809, 2479031, 171572636187431, 3516743833

Offset: 0

N. J. A. Sloane, Dec 03 2008

nonn

From David Pasino, Dec 03 2008: (Start)
The number of refinements of a partition is the product of the Bell numbers of the cell sizes.
The number of encoarsements is the Bell number of the number of cells.
For these to be equal, a Bell number has to be a product of Bell numbers.
This happens if there are n-1 single-element cells and 1 n-element cell.
Does it ever happen otherwise? (End)

Amiram Eldar, Table of n, a(n) for n = 0..104 (terms 0..70 from T. D. Noe)
Simon Plouffe, Factors of Bell numbers [_David Pasino_, Dec 03 2008]
Prime-Numbers.org, Prime number checker up to 10000000000 [_David Pasino_, Dec 03 2008]

[1,1] cat [Maximum(PrimeDivisors(Bell(n))): n in [2..30]]; // Vincenzo Librandi, Jan 04 2017

Join[{1}, Table[FactorInteger[BellB[n]][[-1, 1]], {n, 40}]] (* Vincenzo Librandi, Jan 04 2017 *)

a(15)-a(20) from David Pasino, Dec 03 2008
a(21) onwards from N. J. A. Sloane, Dec 04 2008
Corrected by David Pasino, Dec 14 2008

A168408 E.g.f.: Sum_{n>=0} (exp(3^n*x) - 1)^n/n!, an analog of the Bell numbers (A000110).

Original entry on oeis.org

1, 3, 90, 21897, 46281375, 882516062106, 153201395082609531, 241604140428719375972139, 3448358784659838019970862469260, 444238039567848645977924129826080612043

Offset: 0

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

nonn

			E.g.f.: A(x) = 1 + 3*x + 90*x^2/2! + 21897*x^3/3! + 46281375*x^4/4! +...
A(x) = 1 + (exp(3*x) - 1) + (exp(9*x) - 1)^2/2! + (exp(27*x) - 1)^3/3! +...+ (exp(3^n*x) - 1)^n/n! +...
a(n) = coefficient of x^n/n! in Bell(x)^(3^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, A168407.

Cf. A008277.

Table[BellB[n, 3^n], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2025 *)

{a(n)=local(infnty=n^5+10);round(exp(-3^n)*sum(k=0,infnty,(3^k*k)^n/k!))}

{a(n)=n!*polcoeff(sum(k=0,n,(exp(3^k*x +x*O(x^n))-1)^k/k!),n)}

{a(n)=n!*polcoeff(exp(3^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)*3^(n*k))} \\ Paul D. Hanna, Feb 15 2010

{a(n)=polcoeff(sum(k=0,n,(3^k*x)^k/prod(j=1,k,1-j*3^k*x+x*O(x^n))),n)}

a(n) = exp(-3^n)*Sum_{k>=0} (3^k*k)^n/k!.
a(n) = [x^n/n! ] Bell(x)^(3^n) where Bell(x) = exp(exp(x) - 1) is the e.g.f. of the Bell numbers.
a(n) = Sum_{k=0..n} S2(n,k)*3^(n*k), where S2(n,k) = A008277(n,k) are the Stirling numbers of the second kind.
G.f.: A(x) = Sum_{n>=0} 3^(n^2)*x^n/[Product_{k=1..n} (1-k*3^n*x)].
a(n) ~ 3^(n^2). - Vaclav Kotesovec, Jul 02 2025

A177255 a(n) = Sum_{j=1..n} j*B(j-1), where B(k) = A000110(k) are the Bell numbers.

Original entry on oeis.org

0, 1, 3, 9, 29, 104, 416, 1837, 8853, 46113, 257583, 1533308, 9676148, 64452909, 451475027, 3314964857, 25442301577, 203604718076, 1695172374548, 14654631691569, 131309475792709, 1217516798735521, 11664652754184043, 115319114738472472, 1174967255260496776

Offset: 0

Emeric Deutsch, May 07 2010

nonn

Number of adjacent blocks in all partitions of the set {1,2,...,n}. An adjacent block is a block of the form (i, i+1, i+2, ...). Example: a(3)=9 because in 1-2-3, 1-23, 12-3, 13-2, and 123 we have 3, 2, 2, 1, and 1 adjacent blocks, respectively.

Harvey P. Dale, Table of n, a(n) for n = 0..575

Cf. A000110, A177254, A177256, A177257.

Partial sums of A052889.

[n eq 0 select 0 else (&+[j*Bell(j-1): j in [1..n]]): n in [0..30]]; // G. C. Greubel, May 11 2024

with(combinat): a := proc (n) options operator, arrow: sum(j*bell(j-1), j = 1 .. n) end proc; seq(a(n), n = 0 .. 23);

With[{nn=30},Join[{0},Accumulate[BellB[Range[0,nn-1]]Range[nn]]]] (* Harvey P. Dale, Nov 10 2014 *)

[sum(j*bell_number(j-1) for j in range(1,1+n)) for n in range(31)] # G. C. Greubel, May 11 2024

a(n) = Sum_{k=0..n} k * A177254(n,k).

A177257 a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*A000110(j).

Original entry on oeis.org

0, 0, 0, 1, 8, 47, 258, 1426, 8154, 48715, 305012, 2001719, 13754692, 98801976, 740584196, 5782218745, 46942426080, 395607218279, 3455493024350, 31236784338746, 291836182128670, 2814329123555051, 27980637362452980

Offset: 0

Emeric Deutsch, May 07 2010

nonn

Number of blocks not consisting of consecutive integers in all partitions of the set {1,2,...,n} (a singleton is considered a block of consecutive integers). Example: a(3)=1 because in 1-2-3, 1-23, 12-3, 13-2, and 123 only the block 13 does not consist of consecutive integers.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000110, A005493, A177254, A177255, A177256.

A177257:= func< n | n eq 0 select 0 else (&+[(Binomial(n,j)-(j+1))*Bell(j): j in [0..n-1]]) >;
[A177257(n): n in [0..30]]; // G. C. Greubel, May 12 2024

with(combinat): a:= proc(n) add((binomial(n, j)-j-1)*bell(j), j = 0 .. n-1) end proc: seq(a(n), n = 0 .. 22);

Table[Sum[(Binomial[n,j]-j-1)BellB[j],{j,0,n-1}],{n,0,30}] (* Harvey P. Dale, Oct 15 2015 *)

def A177257(n): return sum((binomial(n,j) -(j+1))*bell_number(j) for j in range(n))
[A177257(n) for n in range(31)] # G. C. Greubel, May 12 2024

a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*Bell(j), where Bell(n) = A000110(n) are the Bell numbers.
a(n) = Sum_{k=0..floor(n/2)} k*A177256(n,k).
a(n) = A005493(n-1) - A177255(n).

A214810 Triangle read by rows: T(n,k) (n>=1, 0 <= k <= p where p = n-th prime) = Bell(k) mod p (cf. A000110).

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 5, 1, 3, 0, 2, 1, 1, 2, 5, 4, 8, 5, 8, 4, 5, 2, 2, 1, 1, 2, 5, 2, 0, 8, 6, 6, 9, 2, 9, 11, 2, 1, 1, 2, 5, 15, 1, 16, 10, 9, 16, 1, 15, 11, 6, 15, 11, 14, 2, 1, 1, 2, 5, 15, 14, 13, 3, 17, 0, 18, 4, 5, 7, 14, 16, 15, 1, 10, 2, 1, 1, 2, 5, 15, 6, 19, 3, 0, 10, 9, 1, 20, 1, 12, 9, 5, 6, 6, 9, 4, 16, 22, 2, 1, 1, 2, 5, 15, 23, 0

Offset: 1

N. J. A. Sloane, Jul 31 2012

The n-th row gives Bell numbers mod prime(n) and has length prime(n)+1.

			Triangle begins:
  [1, 1, 0],
  [1, 1, 2, 2],
  [1, 1, 2, 0, 0, 2],
  [1, 1, 2, 5, 1, 3, 0, 2],
  [1, 1, 2, 5, 4, 8, 5, 8, 4, 5, 2, 2],
  [1, 1, 2, 5, 2, 0, 8, 6, 6, 9, 2, 9, 11, 2],
  [1, 1, 2, 5, 15, 1, 16, 10, 9, 16, 1, 15, 11, 6, 15, 11, 14, 2],
  [1, 1, 2, 5, 15, 14, 13, 3, 17, 0, 18, 4, 5, 7, 14, 16, 15, 1, 10, 2],
  ...

Alois P. Heinz, Rows n = 1..100, flattened
J. Levine and R. E. Dalton, Minimum Periods, Modulo p, of First Order Bell Exponential Integrals, Mathematics of Computation, 16 (1962), 416-423. See Table 2.

Cf. A000110, A054767.

T:= n-> (p-> seq(combinat[bell](k) mod p, k=0..p))(ithprime(n)):
seq(T(n), n=1..10);  # Alois P. Heinz, Jun 07 2023

A214810row[n_]:=Mod[BellB[Range[0,Prime[n]]],Prime[n]];Array[A214810row,50] (* Paolo Xausa, Aug 07 2023 *)

A224922 O.g.f. satisfies: A(x) = Sum_{n>=0} A000110(n)x^nA(x)^n, where A000110 are the Bell numbers.

Original entry on oeis.org

1, 1, 3, 12, 56, 288, 1586, 9201, 55675, 349159, 2260209, 15063260, 103201968, 726380164, 5252083746, 39029442336, 298340012448, 2348365289852, 19058365017840, 159659978176454, 1382148854813222, 12373696834781918, 114606230432935860, 1098199966940781258

Offset: 0

Paul D. Hanna, Apr 19 2013

nonn

			O.g.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 56*x^4 + 288*x^5 + 1586*x^6 +...
The o.g.f. satisfies:
(1) A(x) = 1 + x*A(x) + 2*x^2*A(x)^2 + 5*x^3*A(x)^3 + 15*x^4*A(x)^4 + 52*x^5*A(x)^5 + 203*x^6*A(x)^6 +...+ A000110(n)*x^n*A(x)^n +...
(2) A(x) = 1 + x*A(x)/(1-x*A(x)) + x^2*A(x)^2/((1-x*A(x))*(1-2*x*A(x))) + x^3*A(x)^3/((1-x*A(x))*(1-2*x*A(x))*(1-3*x*A(x))) + x^4*A(x)^4/((1-x*A(x))*(1-2*x*A(x))*(1-3*x*A(x))*(1-4*x*A(x))) +...

Cf. A099947, A000110.

{a(n)=if(n<0, 0, polcoeff( 1/x*serreverse(x/serlaplace(exp(exp(x+x*O(x^n))-1))), n))}
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*A^m/prod(k=1, m,1-k*x*A +x*O(x^n)) )); polcoeff(A, n)}
for(n=0,30,print1(a(n), ", "))

O.g.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^n / Product_{k=1..n} (1 - k*x*A(x)).
O.g.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - 1*x*A(x)/(1 - x*A(x)/(1 - 2*x*A(x)/(1 - x*A(x)/(1 - 3*x*A(x)/(1 - x*A(x)/(1 - 4*x*A(x)/(1 - ... ))))))))), a continued fraction.
a(n) = [x^n] ( Sum_{k>=0} A000110(k)*x^k )^(n+1) / (n+1).

A305850 Weigh transform of the Bell numbers (A000110).

Original entry on oeis.org

1, 1, 2, 7, 21, 78, 305, 1304, 6007, 29854, 159012, 904986, 5479078, 35150263, 238033523, 1695554145, 12663533586, 98881246850, 805128085616, 6820302066048, 59983405937707, 546690232627480, 5154757226832625, 50208266917662433, 504482106565647708

Offset: 0

Alois P. Heinz, Jun 11 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..576

Cf. A000110, A290351, A305846, A305852.

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(g(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..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[g[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 12 2022, after Alois P. Heinz *)

G.f.: Product_{k>=1} (1+x^k)^Bell(k).

A051131 Prime Bell numbers (A000110).

Original entry on oeis.org

2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837

Offset: 1

Ignacio Larrosa Cañestro

nonn

			Bell(2)=2 and Bell(3)=5 are primes, Bell(4)..Bell(6) are composite, Bell(7)=877 is prime.

E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
The Prime Database, Bell(2841), the next term which is too large to include here.
Eric Weisstein's World of Mathematics, Bell Number

Cf. A000110, A051130.

A070906 Every third Bell number A000110.

Original entry on oeis.org

1, 5, 203, 21147, 4213597, 1382958545, 682076806159, 474869816156751, 445958869294805289, 545717047936059989389, 846749014511809332450147, 1629595892846007606764728147, 3819714729894818339975525681317

Offset: 0

Benoit Cloitre, May 19 2002

Cf. A000110, A020557.

Table[ BellB[3*n], {n, 0, 12}] (* Jean-François Alcover, Dec 13 2012 *)
BellB[3*Range[0,15]] (* Harvey P. Dale, Apr 19 2020 *)

for(n=0,50,print1(round(sum(i=0,1000,i^(3*n)/(i)!)/exp(1)),","))

from itertools import accumulate, islice
def A070906_gen(): # generator of terms
    yield 1
    blist, b = (1,), 1
    while True:
        for _ in range(3):
            blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield b
A070906_list = list(islice(A070906_gen(),30)) # Chai Wah Wu, Jun 22 2022

[bell_number(3*n) for n in range(0, 13)] # Zerinvary Lajos, May 14 2009

a(n) = Bell(3*n) = A000110(3*n). - Vladeta Jovovic, Feb 02 2003
a(n) = exp(-1)*Sum_{k>=0} k^(3n)/k!.
E.g.f.: exp(x*(d_z)^3)*(exp(exp(z)-1)) |_{z=0}, with the derivative operator d_z := d/dz. Adapted from eqs. (14) and (15) of the 1999 C. M. Bender reference given in A000110.
E.g.f.: exp(-1)*Sum_{n>=0} exp(n^3*x)/n!. - Vladeta Jovovic, Aug 24 2006

A094070 a(n) = A000085(n) * A000110(n).

Original entry on oeis.org

1, 4, 20, 150, 1352, 15428, 203464, 3162960, 55405140, 1101298600, 24222234720, 590544046744, 15715973012248, 456341011254560, 14312979247985120, 484253161428902192, 17550722413456774848, 680244627812139042016, 28053748582811428182080, 1228896901162555453603712

Offset: 0

N. J. A. Sloane, May 01 2004

nonn

Coefficients arising in combinatorial field theory.

P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Some useful combinatorial formulas for bosonic operators, J. Math. Phys. 46, 052110 (2005) (6 pages).
P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G E. H. Duchamp, Combinatorial field theories via boson normal ordering, preprint, Apr 27 2004.

P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Combinatorial field theories via boson normal ordering, arXiv:quant-ph/0405103, 2004-2006. The title of this paper in the arXiv was later changed to "Some useful combinatorial formulas for bosonic operators"
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory

Cf. A000085, A005425, A094071, A094072, A094073, A094074.

with(combinat): with(orthopoly): seq((I/sqrt(2))^(n+1)*H(n+1,-I/sqrt(2))*bell(n+1),n=0..17); # Emeric Deutsch, Nov 22 2004

a[n_] := Sum[StirlingS1[n+1, k] 2^k BellB[k, 1/2], {k, 0, n+1}] BellB[n+1];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 07 2018 *)

a(n) = (i/sqrt(2))^(n+1)*H(n+1, -i/sqrt(2))*Bell(n+1), where i=sqrt(-1), H(n, x) are the Hermite polynomials and Bell(n) are the Bell numbers. - Emeric Deutsch, Nov 22 2004

More terms from Ralf Stephan, Oct 14 2004

cp's OEIS Frontend

A144293 Largest prime factor of n-th Bell number A000110(n) (or 1 if A000110(n) = 1).

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A168408 E.g.f.: Sum_{n>=0} (exp(3^n*x) - 1)^n/n!, an analog of the Bell numbers (A000110).

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A177255 a(n) = Sum_{j=1..n} j*B(j-1), where B(k) = A000110(k) are the Bell numbers.

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A177257 a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*A000110(j).

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A214810 Triangle read by rows: T(n,k) (n>=1, 0 <= k <= p where p = n-th prime) = Bell(k) mod p (cf. A000110).

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A224922 O.g.f. satisfies: A(x) = Sum_{n>=0} A000110(n)*x^n*A(x)^n, where A000110 are the Bell numbers.

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PARI

PARI

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A305850 Weigh transform of the Bell numbers (A000110).

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Maple

A224922 O.g.f. satisfies: A(x) = Sum_{n>=0} A000110(n)x^nA(x)^n, where A000110 are the Bell numbers.