A000110 -id:A000110 - cp's OEIS frontend

A290351 Euler transform of the Bell numbers (A000110).

1, 1, 3, 8, 26, 88, 340, 1411, 6417, 31474, 166242, 939646, 5659613, 36158227, 244049562, 1733702757, 12919475840, 100690425442, 818554392962, 6924577964036, 60828588178031, 553821749290234, 5217264062756556, 50776256646839085, 509823607380230570

Offset: 0

Alois P. Heinz, Jul 28 2017

nonn

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

Cf. A000110, A085686, A290352, A305850.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      b(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);

b[n_]:=b[n]=If[n==0, 1, Sum[b[n - j] Binomial[n - 1, j - 1], {j, n}]]; a[n_]:=a[n]=If[n==0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}] a[n - j], {j, n}]/n]; Table[a[n], {n, 0, 50}] (* Indranil Ghosh, Jul 28 2017, after Maple code *)

A327432 Smallest k > 0, such that A000110(k) is divisible by n, if such a number exists, otherwise 0.

Original entry on oeis.org

1, 2, 4, 5, 3, 8, 6, 0, 8, 8, 31, 8, 5, 20, 4, 0, 22, 8, 9, 8, 9, 53, 8, 0, 10, 5, 17, 20, 6, 8, 57, 0, 95, 26, 35, 8, 12, 59, 34, 0, 66, 35, 56, 53, 8, 8, 12, 0, 20, 56, 22, 5, 9, 17, 245, 0, 9, 104, 78, 8, 20, 89, 47, 0, 61, 95, 28, 29, 8, 35, 43, 0, 43, 125

Offset: 1

Vaclav Kotesovec, Sep 10 2019

nonn

a(8*j) = 0.

Vaclav Kotesovec, Table of n, a(n) for n = 1..4500

Cf. A000110, A066562, A325630, A327433.

A361380 Sum over the j-th term of the (n-j)-th inverse binomial transform of the Bell numbers (A000110) for all j in [n].

Original entry on oeis.org

1, 2, 3, 6, 17, 56, 215, 922, 4305, 21894, 119539, 696632, 4314925, 28237146, 194602079, 1407456694, 10649642837, 84100177424, 691474151187, 5907288773554, 52340230286509, 480153099982726, 4553711640946919, 44584683333637168, 450075389309517849

Offset: 0

Alois P. Heinz, Mar 09 2023

nonn

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

Antidiagonal sums of A361781.

Cf. A000110, A059841, A290219, A347420.

a:= n-> add(add(binomial(i, j)*(i-n)^(i-j)*combinat[bell](j), j=0..i), i=0..n):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> add(i!*coeff(series(exp(exp(x)-(n-i)*x-1), x, i+1), x, i), i=0..n):
seq(a(n), n=0..25);
# third Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
    end:
a:= n-> add(b(i, i-n), i=0..n):
seq(a(n), n=0..25);

from math import comb
from sympy import bell
def A361380(n): return sum(comb(i,j)*(i-n)**(i-j)*bell(j) for i in range(n+1) for j in range(i+1)) # Chai Wah Wu, Apr 05 2023

a(n) = Sum_{i=0..n} i! * [x^i] exp(exp(x)-(n-i)*x-1).
a(n) = Sum_{0<=j<=i<=n} binomial(i,j)*(i-n)^(i-j)*Bell(j).
a(n) mod 2 = A059841(n).

A361781 A(n,k) is the n-th term of the k-th inverse binomial transform of the Bell numbers (A000110); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, -1, 1, 5, 1, -2, 2, 1, 15, 1, -3, 5, -3, 4, 52, 1, -4, 10, -13, 7, 11, 203, 1, -5, 17, -35, 36, -10, 41, 877, 1, -6, 26, -75, 127, -101, 31, 162, 4140, 1, -7, 37, -139, 340, -472, 293, -21, 715, 21147, 1, -8, 50, -233, 759, -1573, 1787, -848, 204, 3425, 115975

Offset: 0

Alois P. Heinz, Mar 23 2023

			Square array A(n,k) begins:
    1,   1,   1,    1,     1,      1,       1,       1, ...
    1,   0,  -1,   -2,    -3,     -4,      -5,      -6, ...
    2,   1,   2,    5,    10,     17,      26,      37, ...
    5,   1,  -3,  -13,   -35,    -75,    -139,    -233, ...
   15,   4,   7,   36,   127,    340,     759,    1492, ...
   52,  11, -10, -101,  -472,  -1573,   -4214,   -9685, ...
  203,  41,  31,  293,  1787,   7393,   23711,   63581, ...
  877, 162, -21, -848, -6855, -35178, -134873, -421356, ...

Alois P. Heinz, Antidiagonals n = 0..150, flattened

Columns k=0-5 give: A000110, A000296, A126617, A346738, A346739, A346740.
Rows n=0-2 give: A000012, A024000, A160457.
Main diagonal gives A290219.
Antidiagonal sums give A361380.
Cf. A108087.

T:= func< n,k | (&+[(-k)^j*Binomial(n,j)*Bell(n-j): j in [0..n]]) >;
A361781:= func< n,k | T(k, n-k) >;
[A361781(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 12 2024

A:= proc(n, k) option remember; uses combinat;
      add(binomial(n, j)*(-k)^j*bell(n-j), j=0..n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
    end:
A:= (n, k)-> b(n, -k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

T[n_, k_]:= T[n, k]= If[k==0, BellB[n], Sum[(-k)^j*Binomial[n,j]*BellB[n-j], {j,0,n}]];
A361781[n_, k_]= T[k, n-k];
Table[A361781[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 12 2024 *)

def T(n,k): return sum( (-k)^j*binomial(n,j)*bell_number(n-j) for j in range(n+1))
def A361781(n, k): return T(k, n-k)
flatten([[A361781(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 12 2024

E.g.f. of column k: exp(exp(x) - k*x - 1).

A(n,k) = Sum_{j=0..n} (-k)^j*binomial(n,j)*Bell(n-j).

A007311 Reversion of o.g.f. for Bell numbers (A000110) omitting a(0)=1.

Original entry on oeis.org

1, -2, 3, -5, 7, -14, 11, -66, -127, -992, -5029, -30899, -193321, -1285300, -8942561, -65113125, -494605857, -3911658640, -32145949441, -274036507173, -2419502677445, -22093077575496, -208364964369913, -2027216779571754, -20323053380033763, -209715614081160850

Offset: 1

N. J. A. Sloane, Mira Bernstein

sign

As the definition says, this entry deliberately omits the zero-th term 1. - N. J. A. Sloane, Jun 16 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Index entries for reversions of series

Cf. A000110.

read transforms; A := series(exp(exp(x)-1),x,60); SERIESTOLISTMULT(%); subsop(1=NULL,%); REVERT(%);
# Alternative, using function CompInv from A357588:
CompInv(26, n -> combinat:-bell(n)); # Peter Luschny, Oct 05 2022

a(n)=if(n<1,0,polcoeff(serreverse(-1+serlaplace(exp(exp(x+x*O(x^n))-1))),n))

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} Bell(k) * A(x)^k. - Ilya Gutkovskiy, Apr 22 2020

Signs corrected Dec 24 2001

A051140 a(n) = (A000110(n) - A000994(n+2))/2.

Original entry on oeis.org

0, 0, 0, 0, 1, 8, 47, 259, 1437, 8208, 48659, 300553, 1937958, 13052028, 91784452, 673328720, 5146726651, 40934788112, 338292605257, 2900716040495, 25769979503573, 236876549095324, 2249873611378525, 22053647698650293, 222832106687369092

Offset: 0

N. J. A. Sloane

nonn

			a(7) = (877 - 359)/2 = 259.

Cf. A000110, A000994.

A061462 The exact power of 2 that divides the n-th Bell number (A000110). Has period 12.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 4, 1, 1, 2

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 10 2001

{ Bell(n) mod 8 } is periodic with period 24, the period being (1 1 2 5 7 4 3 5 4 3 7 2 5 5 2 1 3 4 7 1 4 7 3 2). Hence the highest power of 2 dividing a Bell number is 4. - David W. Wilson, Jun 29 2001

Amiram Eldar, Table of n, a(n) for n = 0..10000
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000110.

PadRight[{},120,{1,1,2,1,1,4,1,1,4,1,1,2}] (* Harvey P. Dale, Sep 24 2017 *)

a(n)=[1, 1, 2, 1, 1, 4, 1, 1, 4, 1, 1, 2][n%12+1] \\ Charles R Greathouse IV, Jul 13 2016

A068939 a(n) = Bell(n^2), where Bell(n) are the Bell numbers, cf. A000110.

Original entry on oeis.org

1, 1, 15, 21147, 10480142147, 4638590332229999353, 3819714729894818339975525681317, 10726137154573358400342215518590002633917247281

Offset: 0

Karol A. Penson, Mar 08 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..20

Cf. A000110.

[Bell(n^2): n in [0..8]]; // Vincenzo Librandi, Jul 16 2013

Table[BellB[n^2], {n, 0, 8}] (* Vincenzo Librandi, Jul 16 2013 *)

for(n=0,50,print1(round(sum(i=0,1000,i^(n^2)/(i)!)/exp(1)),","))

from sympy import bell
def A068939(n): return bell(n**2) # Chai Wah Wu, Jun 22 2022

a(n) = exp(-1)*Sum(k=>0, k^(n^2)/k!). - Benoit Cloitre, May 19 2002

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

A092460 Numbers that are not Bell numbers (A000110).

Original entry on oeis.org

0, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

N. J. A. Sloane, Mar 26 2004

nonn

With[{b=BellB[Range[6]]},Complement[Range[0,Last[b]],b]] (* Harvey P. Dale, Sep 05 2020 *)

from itertools import accumulate, islice
def A092460_gen(): # generator of terms
    yield 0
    blist, b = (1,), 1
    while True:
        blist = list(accumulate(blist, initial=b))
        yield from range(b+1,b:=blist[-1])
A092460_list = list(islice(A092460_gen(),30)) # Chai Wah Wu, Jun 22 2022

A113883 Indices of semiprime Bell numbers A000110.

Original entry on oeis.org

4, 6, 16, 31, 33, 49, 84

Offset: 1

Jonathan Vos Post, Jan 27 2006

Semiprime analog of A051130 (indices of prime Bell numbers). These indices of semiprime Bell numbers include all values through B(120), which has 146 digits and at least 5 prime factors, the smallest being 71.

			a(1) = 4 because B(4) = 15 = 3 * 5.
a(2) = 6 because B(6) = 203 = 7 * 29.
a(3) = 16 because B(16) = 10480142147 = 241 * 43486067.
a(4) = 31 because B(31) = 10293358946226376485095653
= 11 * 935759904202397862281423.
a(5) = 33 because B(33) = 1629595892846007606764728147
= 5694673 * 286161451736738458339.
a(6) = 49 because B(49) =
10726137154573358400342215518590002633917247281
= 7615441337805454611187 *
1408472165798904899327563.
a(7) = 84 because B(84) is a 93-digit semiprime, whose smaller prime factor is 8429925224798761223.

John Sokol, The First 1000 Bell Numbers.

Cf. A000110, A001358, A113015.

n such that A000110(n) is semiprime. n such that A000110(n) is in A001358.

cp's OEIS Frontend

A290351 Euler transform of the Bell numbers (A000110).

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A327432 Smallest k > 0, such that A000110(k) is divisible by n, if such a number exists, otherwise 0.

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A361380 Sum over the j-th term of the (n-j)-th inverse binomial transform of the Bell numbers (A000110) for all j in [n].

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A361781 A(n,k) is the n-th term of the k-th inverse binomial transform of the Bell numbers (A000110); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A007311 Reversion of o.g.f. for Bell numbers (A000110) omitting a(0)=1.

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A051140 a(n) = (A000110(n) - A000994(n+2))/2.

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A061462 The exact power of 2 that divides the n-th Bell number (A000110). Has period 12.

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A068939 a(n) = Bell(n^2), where Bell(n) are the Bell numbers, cf. A000110.

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