A000110 -id:A000110 - cp's OEIS frontend

A070071 a(n) = n*B(n), where B(n) are the Bell numbers, A000110.

0, 1, 4, 15, 60, 260, 1218, 6139, 33120, 190323, 1159750, 7464270, 50563164, 359377681, 2672590508, 20744378175, 167682274352, 1408702786668, 12277382510862, 110822101896083, 1034483164707440, 9972266139291771, 99147746245841106, 1015496134666939958

Offset: 0

Karol A. Penson, Apr 19 2002

nonn

a(n) is the total number of successions among all partitions of {1,2,...,n+1}; a succession is a pair (i,i+1) of consecutive integers lying in a block. For example, a(3)=15 because {1,2,3,4} has 6 partitions with 1 succession - 1/2/34, 1/23/4, 12/3/4, 14/23, 134/2, 124/3, 3 partitions with 2 successions - 1/234, 123/4, 12/34 and 1 partition with 3 successions - 1234. Thus a(3) = 6*1 + 3*2 + 1*3 = 15. - Augustine O. Munagi, Jul 01 2008
a(n) is the number of occurrences of integers in a list of all partitions of the set {1,...,n}. For example, the list 123, 1/23, 2/13, 3/12, 1/2/3 of all partitions of the set {1,2,3} requires 15 occurrences of integers each belonging to that set. [From Michael Hardy (hardy(AT)math.umn.edu), Nov 08 2008]
The bijection between the two foregoing characterizations is as follows: Fix x in {1,2,...,n} and associate x with the succession (x,x+1) which appears in some partitions of {1,2,...,n+1}. Replace x,x+1 by x and partition the n-set {1,2,...,x,x+2,...,n+1}, giving B(n) partitions. Thus the succession (x,x+1) occurs among partitions of {1,2,...,n+1} exactly B(n) times. - Augustine O. Munagi, Jun 02 2010

Alois P. Heinz, Table of n, a(n) for n = 0..574 (terms n=0..200 from Vincenzo Librandi)
Augustine O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 1298--1308.

Cf. A000110, A052889, A105479, A105480, A105481, A175757.

Row sums of A270236, A270701, A270702, A286416, A319298, A319375.

[n*Bell(n): n in [0..25]]; // Vincenzo Librandi, Mar 15 2014

with(combinat): a:=n->sum(numbcomb (n,0)*bell(n), j=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Apr 25 2007
with(combinat): a:=n->sum(bell(n), j=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Apr 25 2007
a:=n->sum(sum(Stirling2(n, k), j=1..n), k=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Jun 28 2007

a[n_] := n!*Coefficient[Series[x E^(E^x+x-1), {x, 0, n}], x, n]
Table[Sum[BellB[n, 1], {i, 1, n}], {n, 0, 21}] (* Zerinvary Lajos, Jul 16 2009 *)
Table[n*BellB[n], {n, 0, 20}] (* Vaclav Kotesovec, Mar 13 2014 *)

a(n)=local(t); if(n<0,0,t=exp(x+O(x^n)); n!*polcoeff(x*t*exp(t-1),n))

[bell_number(n)*n for n in range(22) ] # Zerinvary Lajos, Mar 14 2009

E.g.f: x*exp(x)*exp(exp(x)-1).
Sum_{k=1..n} n*binomial(n-1, k-1)*Bell(n-k), n >= 2. - Zerinvary Lajos, Nov 22 2006
a(n) ~ n^(n+1) * exp(n/LambertW(n)-1-n) / (sqrt(1+LambertW(n)) * LambertW(n)^n). - Vaclav Kotesovec, Mar 13 2014
a(n) = Sum_{k=1..n} k * A175757(n,k). - Alois P. Heinz, Mar 03 2020
a(n) = Sum_{j=0..n} n * Stirling2(n,j). - Detlef Meya, Apr 11 2024

A137341 a(n) = n! * A000110(n) where A000110 is the sequence of Bell numbers.

Original entry on oeis.org

1, 1, 4, 30, 360, 6240, 146160, 4420080, 166924800, 7673823360, 420850080000, 27086342976000, 2018319704755200, 172142484203289600, 16642276683198566400, 1808459441303074560000, 219273812138054209536000, 29473992420094651613184000

Offset: 0

David W. K. Yeung, Eric L. H. Ku and Patricia M. Yeung (wkyeung(AT)hkbu.edu.hk), Apr 08 2008

nonn

Cooperative games are frequently formulated in terms of partition functions. In particular, the set of players may be divided into various coalitions forming partitions with different coalition structures. This recursive sequence identifies the number of partitions in an n-player game where the position of the individual player counts.

Lists of sublists of total size n with up to n different 1s, up to n-1 different 2s, ... generated by successive insertion. Sublists stay ordered as inserted. See example field for illustration. - Olivier Gérard, Aug 12 2016

			a(0) = 1;
a(1) = C(0,0)*a(0)*1!/0! = 1;
a(2) = C(1,1)*a(1)*2!/1! + C(1,0)*a(0)*2!/0! = 4;
a(3) = C(2,2)*a(2)*3!/2! + C(2,1)*a(1)*3!/1! + C(2,0)*a(0)*3!/0! = 30;
a(4) = C(3,3)*a(3)*4!/3! + C(3,2)*a(2)*4!/2! + C(3,1)*a(1)*4!/1! + C(3,0)*a(0)*4!/0! = 360.
From _Olivier Gérard_, Aug 12 2016: (Start)
Illustration as family of lists of sublists extending set partitions.
In this interpretation the lowercase letters allow us to distinguish between integers introduced at each iteration (or generation).
Construction from the family of size n to family of size n+1 is done by insertion.
Insertion is only possible at the end of a sublist or to create a new singleton sublist at the end of the list.
:
1: {{1a}}*
4: {{1a},{1b}}  {{1a,1b}}  {{1a,2b}}*  {{1a},{2b}}*
30: {{1a,1c},{1b}}     {{1a},{1b,1c}}     {{1a},{1b},{1c}}
....{{1a,2c},{1b}}     {{1a},{1b,2c}}     {{1a},{1b},{2c}}
....{{1a,3c},{1b}}     {{1a},{1b,3c}}     {{1a},{1b},{3c}}
....{{1a,1b,1c}}       {{1a,1b},{1c}}
....{{1a,1b,2c}}       {{1a,1b},{2c}}
....{{1a,1b,3c}}       {{1a,1b},{3c}}
....{{1a,2b,1c}}       {{1a,2b,2c}}       {{1a,2b,3c}}*
....{{1a,2b},{1c}}     {{1a,2b},{2c}}     {{1a,2b},{3c}}*
....{{1a,1c},{2b}}     {{1a},{2b,1c}}     {{1a},{2b},{1c}}
....{{1a,2c},{2b}}     {{1a},{2b,2c}}     {{1a},{2b},{2c}}
....{{1a,3c},{2b}}*    {{1a},{2b,3c}}*    {{1a},{2b},{3c}}*
:
The lists of sublists marked with * correspond to classical set partitions counted by Bell numbers A000110. (End)

W. Lucas and R. Thrall, N-person games in partition function form, Naval Research Logistics Quarterly X, pp. 281-298, 1963.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
Abel Lacabanne, Framization of Schur-Weyl duality and Yokonuma-Hecke type algebras, arXiv:2312.14796 [math.RT], 2023. See p. 34.
David W. K. Yeung, Eric L. H. Ku and Patricia M. Yeung, A Recursive Sequence for the Number of Positioned Partition, International Journal of Algebra, Vol. 2 (2008), No. 4, pp. 181-185.

Cf. A000110, A000142.
Cf. A048800 = n!*A000110(n-1).
Main diagonal of A323099 and of A323128.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 30 2019

Table[n!*BellB[n],{n,0,20}] (* Vaclav Kotesovec, Mar 13 2014 *)

Vec(serlaplace(serlaplace(exp(exp(O(x^20)+x)-1)))) \\ Joerg Arndt, Mar 13 2009

from sympy import bell, factorial
[factorial(n) * bell(n) for n in range(101)] # Indranil Ghosh, Mar 20 2017

[factorial(m) * bell_number(m) for m in range(17)]  # Zerinvary Lajos, Jul 06 2008

a(0) = 1; for n>0, a(n) = Sum_{j=0..n-1} binomial(n-1,j) * a(j) * n!/j!.
From the recurrence it follows that A'(x) = exp(x) * A(x) where A(x) = Sum_{k>=0} a(k) * x^k / k!^2. The solution to this differential equation is A(x) = exp(exp(x)). The expression in Joerg Arndt's PARI program follows from this. - Max Alekseyev, Mar 11 2009
a(n) ~ n^(2*n+1/2) * exp(n/LambertW(n)-1-2*n) * sqrt(2*Pi/(1+LambertW(n))) / LambertW(n)^n. - Vaclav Kotesovec, Mar 13 2014
a(n) = n! * Sum_{j=0..n} Stirling2(n,j). - Detlef Meya, Apr 11 2024

Edited by N. J. A. Sloane, Sep 19 2009

More terms from Vincenzo Librandi, Mar 15 2014

A105479 a(n) = C(n,2)*Bell(n-2) (cf. A000217, A000110).

Original entry on oeis.org

0, 0, 1, 3, 12, 50, 225, 1092, 5684, 31572, 186300, 1163085, 7654350, 52928460, 383437327, 2902665885, 22907918640, 188082362120, 1603461748491, 14169892736484, 129594593170210, 1224875863061970, 11948280552370932, 120142063487658003, 1243853543811461148

Offset: 0

Augustine O. Munagi, Apr 10 2005

Number of blocks of size 2 in all set partitions of {1,2,...,n}. Example: a(3)=3 because the set partitions of {1,2,3} are 1|2|3, 1|23, 12|3, 13|2 and 123, containing exactly 3 blocks of size 2. a(n) = Sum_{k>=0} k*A124498(n-1,k). - Emeric Deutsch, Nov 06 2006
Number of partitions of {1...n} containing 2 pairs of consecutive integers, where each pair is counted within a block and a string of more than 2 consecutive integers are counted two at a time. E.g. a(4) = 3 because the partitions of {1,2,3,4} with 2 pairs of consecutive integers are 123/4,12/34,1/234. - Augustine O. Munagi, Apr 10 2005
a(n) is the total sum of singletons in all set partitions of [n-1]. a(4) = 12 = 0+1+2+3+6: 123, 1|23, 13|2, 12|3, 1|2|3. - Alois P. Heinz, Mar 06 2024

Augustine O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463.

Cf. A105480, A105489, A105484, A124498.

Column k=2 of A193297.

[seq(binomial(n,2)*combinat[bell](n-2),n=0..50)];

Join[{0,0},Table[Binomial[n,2]BellB[n-2],{n,2,30}]] (* Harvey P. Dale, May 06 2014 *)

from itertools import count, accumulate, islice
def A105479_gen(): # generator of terms
    yield from (0,0,1)
    blist, b, c = (1,), 1, 1
    for n in count(2):
        c += n
        blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield b*c
A105479_list = list(islice(A105479_gen(),20)) # Chai Wah Wu, Jun 22 2022

a(n) = binomial(n-1, 2)*Bell(n-3), the case r = 2 of the general case of r pairs: c(n, r) = binomial(n-1, r)*Bell(n-r-1).
E.g.f.: z^2/2 * e^(e^z-1) - Frank Ruskey, Dec 26 2006
G.f.: exp(-1)*Sum_{n>=0} (x^2/(n!*(1-n*x)^3)). - Vladeta Jovovic, Feb 05 2008
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i<=j), and A[i,j]=0 otherwise. Then, for n>=2, a(n)=(-1)^(n-2)coeff(charpoly(A,x),x^2). - Milan Janjic, Jul 08 2010
G.f.: x^2/exp(1)*G(0), where G(k) = 1 + (2*k*x-1)^3/((2*k+1)*(2*k*x+x-1)^3 - (2*k+1)*(2*k*x+x-1)^6/((2*k*x+x-1)^3 + 2*(k+1)*(2*k*x+2*x-1)^3/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013

Edited by N. J. A. Sloane, Jan 01 2007

A320956 a(n) = A000110(n) * A000111(n). The exponential limit of sec + tan. Row sums of A373428.

Original entry on oeis.org

1, 1, 2, 10, 75, 832, 12383, 238544, 5733900, 167822592, 5859172975, 240072637440, 11388362495705, 618357843791872, 38057876106154882, 2632817442236631040, 203225803724876875315, 17390464322078045896704, 1640312648221489789841119, 169667967895669459925991424

Offset: 0

Peter Luschny, Nov 07 2018

nonn

We say that the sequence S is the exponential limit of the function f relative to the kernel K if and only if the exponential generating functions
  egf(n) = Sum_{k=0..n} K(n, k)*f(x*(n-k)) generate a family of sequences
T(n) = k -> (k!/n!)*[x^k] egf(n) which converge to S. Convergence here means that for every fixed k the terms T(n)(k) differ from S(k) only for finitely many indices.
The paradigmatic example is to set f(x) = exp(x), K(n, k) = !k*binomial(n, k) (!n is the subfactorial of n) and obtain for S the Bell numbers. This example is set forth in A320955.
Let D(f)(x) represent the derivative of f(x) with respect to x and (D^(n))(f) the n-th derivative of f. Then the exponential limit of f is B(n)*((D^(n))(f))(0) where B(n) is the n-th Bell number: ExpLim(f) = f(0), (D(f))(0), 2*((D^(2))(f))(0), 5*((D^(3))(f))(0), 15*((D^(4))(f))(0), 52*((D^(5))(f))(0), ... Since exp is a fixed point of D and exp(0) = 1 we have the identity ExpLim(exp)[n] = B(n). Similarly ExpLim(sin)[n] = B(n)*mod(n,2)*(-1)^binomial(n,2).
If we set f = sec + tan and K(n, k) = !k*binomial(n, k) the exponential limit is this sequence, a(n).

			Illustration of the convergence:
  [0] 1, 0, 0,  0,  0,   0,     0,      0,       0, ... A000007
  [1] 1, 1, 1,  2,  5,  16,    61,    272,    1385, ... A000111
  [2] 1, 1, 2,  8, 40, 256,  1952,  17408,  177280, ... A000828
  [3] 1, 1, 2, 10, 70, 656,  7442,  99280, 1515190, ... A320957
  [4] 1, 1, 2, 10, 75, 816, 11407, 194480, 3871075, ... A321394
  [5] 1, 1, 2, 10, 75, 832, 12322, 232560, 5325325, ...
  [6] 1, 1, 2, 10, 75, 832, 12383, 238272, 5693735, ...
  [7] 1, 1, 2, 10, 75, 832, 12383, 238544, 5732515, ...
  [8] 1, 1, 2, 10, 75, 832, 12383, 238544, 5733900, ...

Peter Luschny, Table of n, a(n) for n = 0..296

Cf. A000111 (n=1), A000828 (n=2), A320957 (n=3), A321394 (n=4).
Cf. A320955 (exp), A320962 (log(x+1)), this sequence (sec+tan), A320958 (arcsin), A320959 (arctanh).
Cf. A373428.

ExpLim := proc(len, f) local kernel, sf, egf:
sf := proc(n) option remember; `if`(n <= 1, 1 - n, (n-1)*(sf(n-1) + sf(n-2))) end:
kernel := proc(n, k) option remember; binomial(n, k)*sf(k) end:
egf := n -> add(kernel(n, k)*f(x*(n-k)), k=0..n):
series(egf(len), x, len+2): seq(coeff(%, x, k)*k!/len!, k=0..len) end:
ExpLim(19, sec + tan);
# Alternative:
explim := (len, f) -> seq(combinat:-bell(n)*((D@@n)(f))(0), n=0..len):
explim(19, sec + tan);
# Or:
a := n -> A000110(n)*A000111(n): seq(a(n), n = 0..19);  # Peter Luschny, Jun 07 2024

m = 20; CoefficientList[Sec[x] + Tan[x] + O[x]^m, x] * Range[0, m-1]! *
BellB[Range[0, m-1]] (* Jean-François Alcover, Jun 19 2019 *)

Name extended by Peter Luschny, Jun 07 2024

A126348 Limit of reversed rows of triangle A126347, in which row sums equal Bell numbers (A000110).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 20, 33, 53, 84, 131, 202, 308, 465, 695, 1030, 1514, 2209, 3201, 4609, 6596, 9386, 13284, 18705, 26211, 36561, 50776, 70226, 96742, 132765, 181540, 247369, 335940, 454756, 613689, 825698, 1107755, 1482038, 1977465, 2631664

Offset: 0

Paul D. Hanna, Dec 31 2006

nonn

In triangle A126347, row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

Row sums of A253830. a(n) equals the number of colored compositions of n, as defined in A253830, whose associated color partition has distinct parts. An example is given below. - Peter Bala, Jan 20 2015

			a(5) = 12: The colored compositions (defined in A253830) of 5 whose color partitions have distinct parts are
5(c1), 5(c2), 5(c3), 5(c4), 5(c5),
1(c1) + 4(c2), 1(c1) + 4(c3), 1(c1) + 4(c4),
3(c1) + 2(c2),
2(c1) + 3(c2), 2(c1) + 3(c3), 2(c2) + 3(c3). - _Peter Bala_, Jan 20 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..500 from Seiichi Manyama)

Cf. A126347, A126349; factorial variant: A126471. A253830, A307599, A307601, A307602.

nmax = 50; CoefficientList[Series[Product[(1 - x + x^k)/(1 - x), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2019 *)

{B(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*B(k,q)*q^k))}
{a(n)=Vec(B(n+1,'q)+O('q^(n*(n-1)/2+1)))[n*(n-1)/2+1]}

{a(n) = local(t); if( n<0, 0, t = 1; polcoeff( sum(k=1, (sqrtint(8*n + 1) - 1)\2, t = t * x^k / (1 - x) / (1 - x^k) + x * O(x^n), 1), n))} /* Michael Somos, Aug 17 2008 */

1 + Sum_{k>0} x^(k * (k + 1) / 2) / ((1 - x)^k * (1 - x) * (1 - x^2) ... (1 - x^k)). - Michael Somos, Aug 17 2008
G.f.: Product_{k>0} (1+x^k/(1-x)). - Vladeta Jovovic, Oct 05 2008
G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1)/(d*(1 - x)^d)). - Ilya Gutkovskiy, Apr 19 2019

A242171 Least prime divisor of B(n) which does not divide any B(k) with k < n, or 1 if such a primitive prime divisor of B(n) does not exist, where B(n) is the n-th Bell number given by A000110.

Original entry on oeis.org

1, 2, 5, 3, 13, 7, 877, 23, 19, 4639, 22619, 37, 27644437, 1800937, 251, 241, 255755771, 19463, 271, 61, 24709, 17, 89, 123419, 367, 101, 157, 67, 75979, 107, 11, 179167, 5694673, 111509, 980424262253, 193, 44101, 5399, 6353, 3221

Offset: 1

Zhi-Wei Sun, May 06 2014

Conjecture: (i) a(n) > 1 for all n > 1.

Conjecture: (ii) For any integer n > 2, the derangement number D(n) given by A000166 has a prime divisor dividing none of those D(k) with 1 < k < n.

Amiram Eldar, Table of n, a(n) for n = 1..104 (terms 1..81 from Zhi-Wei Sun)

Cf. A000040, A000110, A000166, A242169, A242170, A242173.

a(4) = 3 since B(4) = 3*5 with 3 dividing none of B(1) = 1, B(2) = 2 and B(3) = 5.

b[n_]:=BellB[n]
f[n_]:=FactorInteger[b[n]]
p[n_]:=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
Do[If[b[n]<2,Goto[cc]];Do[Do[If[Mod[b[i],Part[p[n],k]]==0,Goto[aa]],{i,1,n-1}];Print[n," ",Part[p[n],k]];Goto[bb];Label[aa];Continue,{k,1,Length[p[n]]}];Label[cc];Print[n," ",1];Label[bb];Continue,{n,1,40}]

# Python 3.2 or higher required.
from itertools import accumulate
from sympy import primefactors
A242171_list, bell, blist, b = [1], [1,1], [1], 1
for _ in range(20):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    fs = primefactors(b)
    for p in fs:
        if all([n % p for n in bell]):
            A242171_list.append(p)
            break
    else:
        A242171_list.append(1)
    bell.append(b) # Chai Wah Wu, Sep 19 2014

A051130 Indices of prime Bell numbers A000110.

Original entry on oeis.org

2, 3, 7, 13, 42, 55, 2841

Offset: 1

Ignacio Larrosa Cañestro

Bell(2841) has been certified to be a prime using Primo. This took 17 months on a P3-800, a P4-2400 and finally a P4-2800. There are no other terms below 6000. - Ignacio Larrosa Cañestro, Feb 13 2004
The next term, if it exists, is > 50000. - Vaclav Kotesovec, May 18 2021
No other terms < 100000. - Mathieu Gouttenoire, Oct 31 2021

			The Bell numbers Bell(2)=2, Bell(3)=5, Bell(7)=877 etc. are primes.

E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
The Prime Database, 93074010508593618333...(6499 other digits)...83885253703080601131
Eric Weisstein's World of Mathematics, Bell Number
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A000110, A051131.

[n: n in [0..1000]|IsPrime(Bell(n))]; // Vincenzo Librandi, Jan 30 2016

Reap[For[n = 1, n <= 3000, n++, If[PrimeQ[BellB[n]], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jun 05 2012 *)
Select[Range[2900],PrimeQ[BellB[#]]&] (* Harvey P. Dale, Nov 08 2012 *)

A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 2, 3, 5, 15, 5, 7, 10, 15, 52, 15, 20, 27, 37, 52, 203, 52, 67, 87, 114, 151, 203, 877, 203, 255, 322, 409, 523, 674, 877, 4140, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 21147, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147

Offset: 0

Gary W. Adamson, May 30 2004

Or, prefix Aitken's array (A011971) with a leading diagonal of 0's and take the differences of each row to get the new triangle.
With offset 1, triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} in which k is the largest entry in the block containing 1 (1 <= k <= n). - Emeric Deutsch, Oct 29 2006
Row term sums = the Bell numbers starting with A000110(1): 1, 2, 5, 15, ...
The k-th term in the n-th row is the number of permutations of length n starting with k and avoiding the dashed pattern 23-1. Equivalently, the number of permutations of length n ending with k and avoiding 1-32. - Andrew Baxter, Jun 13 2011
From Gus Wiseman, Aug 11 2020: (Start)
Conjecture: Also the number of divisors d with distinct prime multiplicities of the superprimorial A006939(n) that are of the form d = m * 2^k where m is odd. For example, row n = 4 counts the following divisors:
  1     2     4    8     16
  3     18    12   24    48
  5     50    20   40    80
  7     54    28   56    112
  9     1350  108  72    144
  25          540  200   400
  27          756  360   432
  45               504   720
  63               600   1008
  75               1400  1200
  135                    2160
  175                    2800
  189                    3024
  675                    10800
  4725                   75600
Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i whose nonzero values are distinct and such that v_n = k.
Crossrefs:
A008278 is the version counted by omega A001221.
A336420 is the version counted by Omega A001222.
A006939 lists superprimorials or Chernoff numbers.
A008302 counts divisors of superprimorials by Omega.
A022915 counts permutations of prime indices of superprimorials.
A098859 counts partitions with distinct multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
Cf. A000005, A000142, A027423, A076954, A124010, A146291, A181818, A336417, A336419, A336421, A336499, A336942.
(End)

			Triangle starts:
   1;
   1,  1;
   2,  1,  2;
   5,  2,  3,  5;
  15,  5,  7, 10, 15;
  52, 15, 20, 27, 37, 52;
From _Gus Wiseman_, Aug 11 2020: (Start)
Row n = 3 counts the following set partitions (described in Emeric Deutsch's comment above):
  {1}{234}      {12}{34}    {123}{4}    {1234}
  {1}{2}{34}    {12}{3}{4}  {13}{24}    {124}{3}
  {1}{23}{4}                {13}{2}{4}  {134}{2}
  {1}{24}{3}                            {14}{23}
  {1}{2}{3}{4}                          {14}{2}{3}
(End)

Alois P. Heinz, Rows n = 0..150, flattened (first 51 rows from Chai Wah Wu)
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv:1108.2642 [math.CO], 2011.
Anders Claesson, Generalized pattern avoidance, Europ. J. Combin., 22 7 (2001), 961-971. See Proposition 3.
A. Bernini, M. Bouvel and L. Ferrari, Some statistics on permutations avoiding generalized patterns, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237 (see transposed array p. 227).

Cf. A000110, A005493, A008278, A011971, A188919, A271466.

T(2n,n) gives A020556.

with(combinat): T:=proc(n,k) if k=1 then bell(n-1) elif k=2 and n>=2 then bell(n-2) elif k<=n then add(binomial(k-2,i)*bell(n-2-i),i=0..k-2) else 0 fi end: matrix(8,8,T): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
Q[1]:=t*s: for n from 2 to 11 do Q[n]:=expand(t^n*subs(t=1,Q[n-1])+s*diff(Q[n-1],s)-Q[n-1]+s*Q[n-1]) od: for n from 1 to 11 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 11 do seq(coeff(P[n],t,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Oct 29 2006
A011971 := proc(n,k) option remember ; if k = 0 then if n=0 then 1; else A011971(n-1,n-1) ; fi ; else A011971(n,k-1)+A011971(n-1,k-1) ; fi ; end: A000110 := proc(n) option remember; if n<=1 then 1 ; else add( binomial(n-1,i)*A000110(n-1-i),i=0..n-1) ; fi ; end: A095149 := proc(n,k) option remember ; if k = 0 then A000110(n) ; else A011971(n-1,k-1) ; fi ; end: for n from 0 to 11 do for k from 0 to n do printf("%d, ",A095149(n,k)) ; od ; od ; # R. J. Mathar, Feb 05 2007
# alternative Maple program:
b:= proc(n, m, k) option remember; `if`(n=0, 1, add(
      b(n-1, max(j, m), max(k-1, -1)), j=`if`(k=0, m+1, 1..m+1)))
    end:
T:= (n, k)-> b(n, 0, n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 20 2018

nmax = 10; t[n_, 1] = t[n_, n_] = BellB[n-1]; t[n_, 2] = BellB[n-2]; t[n_, k_] /; n >= k >= 3 := t[n, k] = t[n, k-1] + t[n-1, k-1]; Flatten[ Table[ t[n, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 15 2011, from formula with offset 1 *)

# requires Python 3.2 or higher.
from itertools import accumulate
A095149_list, blist = [1,1,1], [1]
for _ in range(2*10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A095149_list += [blist[-1]]+ blist
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

With offset 1, T(n,1) = T(n,n) = T(n+1,2) = B(n-1) = A000110(n-1) (the Bell numbers). T(n,k) = T(n,k-1) + T(n-1,k-1) for n >= k >= 3. T(n,n-1) = B(n-1) - B(n-2) = A005493(n-3) for n >= 3 (B(q) are the Bell numbers A000110). T(n,k) = A011971(n-2,k-2) for n >= k >= 2. In other words, deleting the first row and first column we obtain Aitken's array A011971 (also called Bell or Pierce triangle; offset in A011971 is 0). - Emeric Deutsch, Oct 29 2006

T(n,1) = B(n-1); T(n,2) = B(n-2) for n >= 2; T(n,k) = Sum_{i=0..k-2} binomial(k-2,i)*B(n-2-i) for 3 <= k <= n, where B(q) are the Bell numbers (A000110). Generating polynomial of row n is P[n](t) = Q[n](t,1), where Q[n](t,s) = t^n*Q[n-1](1,s) + s*dQ[n-1](t,s)/ds + (s-1) Q[n-1](t,s); Q[1](t,s) = ts. - Emeric Deutsch, Oct 29 2006

Corrected and extended by R. J. Mathar, Feb 05 2007

Entry revised by N. J. A. Sloane, Jun 01 2005, Jun 16 2007

A101053 a(n) = n! * Sum_{k=0..n} Bell(k)/k! (cf. A000110).

Original entry on oeis.org

1, 2, 6, 23, 107, 587, 3725, 26952, 219756, 1998951, 20105485, 221838905, 2666280457, 34689290378, 485840964614, 7288997427755, 116634438986227, 1982868327635663, 35692311974248093, 678159760252918824, 13563246929216611852, 284828660383365005643

Offset: 0

Karol A. Penson, Nov 29 2004

nonn

Sequence was originally defined as an infinite sum involving generalized Laguerre polynomials: a(n) = ((-1)^n*n!/exp(1))*Sum_{k>=0} LaguerreL(n,-n-1,k)/k!, n=0,1... . It appears in the problem of normal ordering of functions of boson operators.

a(n) is the number of ways to linearly order the elements in a (possibly empty) subset S of {1,2,...,n} and then partition the complement of S. - Geoffrey Critzer, Aug 07 2015

Robert Israel, Table of n, a(n) for n = 0..450

Cf. A000110, A186755, A278677, A352270.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(Exp(x)-1)/(1-x) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Mar 31 2019

with(combinat): a:=n->add(bell(j)*n!/j!,j=0..n): seq(a(n),n=0..20); # Zerinvary Lajos, Mar 19 2007

nn = 21; Range[0, nn]! CoefficientList[Series[Exp[(Exp[x]-1)]/(1-x), {x, 0, nn}], x] (* Geoffrey Critzer, Aug 07 2015 *)

egf(s)=my(v=Vec(s),i); while(polcoeff(s,i)==0,i++); i--; vector(i+#v,j,polcoeff(s,j+i)*(j+i)!)
egf(exp(exp(x)-1)/(1-x)) \\ Charles R Greathouse IV, Aug 07 2015

my(x='x+O('x^30)); Vec(serlaplace( exp(exp(x)-1)/(1-x) )) \\ G. C. Greubel, Mar 31 2019

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, ((j-1)!+1)*binomial(i-1, j-1)*v[i-j+1])); v; \\ Seiichi Manyama, Jul 14 2022

m = 30; T = taylor(exp(exp(x)-1)/(1-x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Mar 31 2019

E.g.f: exp(exp(x)-1)/(1-x).
a(n) ~ exp(exp(1)-1) * n!. - Vaclav Kotesovec, Jun 26 2022
a(0) = 1; a(n) = Sum_{k=1..n} ((k-1)! + 1) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jul 14 2022

New definition from Vladeta Jovovic, Dec 01 2004

A126347 Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)B(k,q)q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 4, 2, 1, 1, 1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1, 1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1, 1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1, 1, 7, 21, 56, 105, 161, 231, 302, 356, 379, 392, 384, 358, 314, 262

Offset: 0

Paul D. Hanna, Dec 31 2006, May 28 2007

Limit of reversed rows equals A126348. Largest term in rows equal A126349.

			Number of terms in row n is: n*(n-1)/2 + 1.
Row functions B(n,q) begin:
  B(0,q) = 1;
  B(1,q) = 1;
  B(2,q) = 1 + q;
  B(3,q) = 1 + 2*q + q^2 + q^3;
  B(4,q) = 1 + 3*q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + q^6.
Triangle begins:
  1;
  1;
  1, 1;
  1, 2, 1, 1;
  1, 3, 3, 4, 2, 1, 1;
  1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1;
  1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1;
  1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..60, flattened
Carl G. Wagner, Partition Statistics and q-Bell Numbers (q = -1), J. Integer Seqs., Vol. 7, 2004.

Row sums give A000110.
Cf. A126348, A126349; factorial variant: A126470.
Cf. A346772.

b:= proc(n, m, t) option remember; `if`(n=0, x^t,
      add(b(n-1, max(m, j), t+j) , j=1..m+1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=n..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Aug 02 2021

B[0, ] = 1; B[n, q_] := B[n, q] = Sum[Binomial[n-1, k] B[k, q] q^k, {k, 0, n-1}] // Expand; Table[CoefficientList[B[n, q], q], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 08 2016 *)

{B(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*B(k,q)*q^k))}
row(n)={Vec(B(n, 'q)+O('q^(n*(n-1)/2+1)))}

/* Alternative formula for the n-th q-Bell number (row n): */ {B(n,q)=local(inf=100);round((0^n + sum(k=1, inf,((q^k-1)/(q-1))^n/prod(i=1,k,(q^i-1)/(q-1)))) / prod(k=1, inf,1 + (q-1)/q^k))}

G.f. for row n: B(n,q) = 1/E_q*{0^n + Sum_{k>=1} [(q^k-1)/(q-1)]^n / q-Factorial(k)}, where q-Factorial(k) = Product_{j=1..k} [(q^j-1)/(q-1)] and where E_q = Sum_{n>=0} 1/q-Factorial(n) = Product_{n>=1} (1+(q-1)/q^n).
Sum_{k=0..n*(n-1)/2} (n+k) * T(n,k) = A346772(n). - Alois P. Heinz, Aug 02 2021
Conjecture: R(n,n) is the (n+1)-th reversed row polynomial where R(0,0) = 1, R(n,k) = R(n-1,n-1) + x^n * Sum_{j=0..k-1} R(n-1,j) for 0 <= k <= n. - Mikhail Kurkov, Jul 06 2025

Keyword:tabl changed to tabf by R. J. Mathar, Oct 21 2010

cp's OEIS Frontend

A070071 a(n) = n*B(n), where B(n) are the Bell numbers, A000110.

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A137341 a(n) = n! * A000110(n) where A000110 is the sequence of Bell numbers.

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A105479 a(n) = C(n,2)*Bell(n-2) (cf. A000217, A000110).

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A320956 a(n) = A000110(n) * A000111(n). The exponential limit of sec + tan. Row sums of A373428.

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A126348 Limit of reversed rows of triangle A126347, in which row sums equal Bell numbers (A000110).

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A242171 Least prime divisor of B(n) which does not divide any B(k) with k < n, or 1 if such a primitive prime divisor of B(n) does not exist, where B(n) is the n-th Bell number given by A000110.

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A126347 Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)B(k,q)q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).