A000110 -id:A000110 - cp's OEIS frontend

A054767 Period of the sequence of Bell numbers A000110 (mod n).

1, 3, 13, 12, 781, 39, 137257, 24, 39, 2343, 28531167061, 156, 25239592216021, 411771, 10153, 48, 51702516367896047761, 39, 109912203092239643840221, 9372, 1784341, 85593501183, 949112181811268728834319677753, 312, 3905, 75718776648063, 117, 1647084

Offset: 1

Eric W. Weisstein, Feb 09 2002

For p prime, a(p) divides (p^p-1)/(p-1) = A023037(p), with equality at least for p up to 19.

Wagstaff shows that N(p) = (p^p-1)/(p-1) is the period for all primes p < 102 and for primes p = 113, 163, 167 and 173. For p = 7547, N(p) is a probable prime, which means that this p may have the maximum possible period N(p) also. See A088790. - T. D. Noe, Dec 17 2008

Jianing Song, Table of n, a(n) for n = 1..102 (b-file based on the Wagstaff article)
J. Levine and R. E. Dalton, Minimum Periods, Modulo p, of First Order Bell Exponential Integrals, Mathematics of Computation, 16 (1962), 416-423.
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979), pp. 1-16.
Jianing Song, A summary of several proofs in the Lunnon article.
Samuel S. Wagstaff Jr., Aurifeuillian factorizations and the period of the Bell numbers modulo a prime, Math. Comp. 65 (1996), 383-391.
Eric Weisstein's World of Mathematics, Bell Number.

Cf. A000110, A023037, A214810. A146093-A146122 gives Bell numbers read mod 3 to mod 32.

(* Warning: this program is just a verification of the existing data
 and should not be used to extend the sequence beyond a(28) *)
BellMod[k_, m_] := Mod[Sum[Mod[StirlingS2[k, j], m], {j, 1, k}], m];
BellMod[k_, 1] := BellB[k];
period[nn_List] := Module[{lgmin=2, lgmax=5, nn1},
   lg=If[Length[nn]<=lgmax, lgmin, lgmax];
   nn1 = nn[[1;;lg]];
   km=Length[nn]-lg;
   Catch[Do[If[nn1==nn[[k;;k+lg-1]], Throw[k-1]];
   If[k==km, Throw[0]], {k, 2, km}]]];
dd[n_] := SelectFirst[Table[{d, n/d},
     {d, Divisors[n][[2;;-2]]}], GCD@@#==1&];
a[1]=1;
a[p_?PrimeQ] := a[p] = (p^p-1)/(p-1);
a[n_/;n>4 && dd[n]!={}] := With[{g = dd[n]}, LCM[a[g[[1]]], a[g[[2]]]]];
a[n_/;MemberQ[FactorInteger[n][[All, 2]], 1]] := a[n]=
   With[{pp = Select[FactorInteger[n], #1[[2]] ==1 &][[All, 1]]},
      a[n/Times@@pp]*Times@@a/@pp];
a[n_/;n>4 && GCD @@ FactorInteger[n][[All, 2]]>1] := a[n]=
   With[{g=GCD @@ FactorInteger[n][[All, 2]]}, n^(1/g)*a[n^(1-1/g)]];
a[n_] := period[Table[BellMod[k, n], {k, 1, 28}]];
Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Jul 31 2012, updated May 06 2024 *)

If gcd(n,m) = 1, a(n*m) = lcm(a(n), a(m)). But the sequence is not in general multiplicative; e.g. a(2) = 3, a(9) = 39 and a(18) = 39. - Franklin T. Adams-Watters, Jun 06 2006

a(2^s) = 3*2^s for s >= 2 (Theorem 6.4 in the Lunnon article). For an odd prime p, if a(p) = (p^p-1)/(p-1) (which is conjectured to hold for all p), then a(p^s) = p^(s-1)*(p^p-1)/(p-1) (Theorem 6.2 in the Lunnon article). - Jianing Song, Jun 18 2025

More information from Phil Carmody, Dec 22 2002
Extended by T. D. Noe, Dec 18 2008
a(26) corrected by Jean-François Alcover, Jul 31 2012
a(18) corrected by Charles R Greathouse IV, Jul 31 2012
a(27)-a(28) from Charles R Greathouse IV, Sep 07 2016

A146093 Bell numbers (A000110) read mod 3.

Original entry on oeis.org

1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1

Offset: 0

N. J. A. Sloane, Feb 07 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..2500
W. F. Lunnon, P. A. B. Pleasants, and N. M. Stephens, Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979), pp. 1-16.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000110, A146094-A146122. A054767 gives periods. See also A002034.

[Bell(n) mod 3: n in [0..100]]; // Vincenzo Librandi, Jan 30 2016

Mod[BellB[Range[0,110]],3] (* Harvey P. Dale, Oct 31 2013 *)

a(n+13) = a(n). - G. C. Greubel, Jan 30 2016

A146122 Bell numbers (A000110) read mod 32.

Original entry on oeis.org

1, 1, 2, 5, 15, 20, 11, 13, 12, 27, 7, 10, 29, 21, 26, 17, 3, 12, 15, 1, 12, 15, 27, 26, 9, 25, 18, 13, 7, 4, 3, 5, 12, 19, 31, 10, 5, 13, 10, 25, 27, 28, 7, 25, 12, 7, 19, 26, 17, 17, 2, 21, 31, 20, 27, 29, 12, 11, 23, 10, 13, 5, 26, 1, 19, 12, 31, 17, 12, 31, 11, 26, 25, 9, 18, 29, 23

Offset: 0

N. J. A. Sloane, Feb 07 2009

G. C. Greubel, Table of n, a(n) for n = 0..10000
W. F. Lunnon, P. A. B. Pleasants, and N. M. Stephens, Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979), pp. 1-16.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A000110, A146093-A146121 (Bell numbers read mod 3 to mod 31).

Cf. A054767 (periods).

[Bell(n) mod 32: n in [0..100]]; // G. C. Greubel, Feb 02 2016

Mod[BellB[Range[0, 100]], 32] (* G. C. Greubel, Feb 02 2016 *)

a(n) = a(n-96). [See A054767. - Jianing Song, Jun 20 2025]

A303601 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + Bell(k) + Bell(m) with 0 <= a <= b and 0 < k <= m, where Bell(k) denotes the k-th Bell number A000110(k).

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 4, 5, 5, 6, 4, 5, 7, 5, 4, 7, 7, 7, 8, 8, 5, 9, 10, 7, 6, 9, 8, 8, 6, 7, 10, 10, 9, 8, 7, 8, 9, 10, 6, 9, 11, 7, 6, 8, 9, 10, 7, 10, 8, 7, 8, 10, 10, 9, 10, 8, 9, 13, 14, 10, 11, 12, 12, 9, 9, 12, 11, 13, 11, 9

Offset: 1

Zhi-Wei Sun, Apr 26 2018

nonn

Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be expressed as the sum of two triangular numbers and two Bell numbers.
This has been verified for all n = 2..7*10^8. Note that 111277 cannot be written as the sum of two squares and two Bell numbers.
As log(Bell(n)) is asymptotically equivalent to n*log(n), Bell numbers eventually grow faster than any exponential function.
See also A303389, A303540, A303543 and A303637 for similar conjectures.

			a(2) = 1 with 2 = 0*(0+1)/2 + 0*(0+1)/2 + Bell(1) + Bell(1).
a(3) = 2 with 3 =  0*(0+1)/2 + 1*(1+1)/2 + Bell(1) + Bell(1) = 0*(0+1)/2 + 0*(0+1)/2 + Bell(1) + Bell(2).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Cf. A000110, A000217, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303637.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
b[n_]:=b[n]=BellB[n];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;k=1;Label[bb];If[b[k]>n,Goto[aa]];Do[If[QQ[4(n-b[k]-b[j])+1],Do[If[TQ[n-b[k]-b[j]-x(x+1)/2],r=r+1],{x,0,(Sqrt[4(n-b[k]-b[j])+1]-1)/2}]],{j,1,k}];k=k+1;Goto[bb];Label[aa];
tab=Append[tab,r],{n,1,70}];Print[tab]

A005001 a(n) = Sum_{k=0..n-1} Bell(k), where the Bell numbers Bell(k) are given in A000110.

Original entry on oeis.org

0, 1, 2, 4, 9, 24, 76, 279, 1156, 5296, 26443, 142418, 820988, 5034585, 32679022, 223578344, 1606536889, 12086679036, 94951548840, 777028354999, 6609770560056, 58333928795428, 533203744952179, 5039919483399502, 49191925338483848, 495150794633289137

Offset: 0

N. J. A. Sloane

Counts rhyme schemes.
Row sums of triangle A137596 starting with offset 1. - Gary W. Adamson, Jan 29 2008
With offset 1 = binomial transform of the Bell numbers, A000110 starting (1, 1, 1, 2, 5, 15, 52, 203, ...). - Gary W. Adamson, Dec 04 2008
a(n) is the number of partitions of the set {1,2,...,n} in which n is either a singleton or it is in a block of consecutive integers. Example: a(3)=4 because we have 123, 1-23, 12-3, and 1-2-3. Deleting the blocks containing n=3, we obtain: empty, 1, 12, 1-2, i.e., all the partitions of the sets: empty, {1}, and {1,2}. - Emeric Deutsch, May 01 2010

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

T. D. Noe, Table of n, a(n) for n = 0..100
J. Riordan, Cached copy of paper
J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.

Partial sums of A000110, partial sums give A029761.
Equals A024716(n-1) + 1.
Cf. A102735, A094262, A000110, A008277, A102639, A003422, A000166, A000204, A000045, A000108.
Cf. A137596.
Cf. A171859.

with(combinat): seq(add(bell(j), j = 0 .. n-1), n = 0 .. 22); # Emeric Deutsch, May 01 2010

nn=20;Range[0,nn]!CoefficientList[Series[Exp[-1](-Exp[Exp[x]]+Exp[1+x]-Exp[x]ExpIntegralEi[1]+Exp[x]ExpIntegralEi[Exp[x]]),{x,0,nn}],x] (* Geoffrey Critzer, Feb 04 2014 *)
BellB /@ Range[0, 30] // Accumulate // Prepend[#, 0]& (* Jean-François Alcover, Oct 19 2019 *)

# Python 3.2 or higher required.
from itertools import accumulate
A005001_list, blist, a, b = [0,1,2], [1], 2, 1
for _ in range(30):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    a += b
    A005001_list.append(a) # Chai Wah Wu, Sep 19 2014

a(0) = 0; for n >= 0, a(n+1) = 1 + Sum_{j=1..n} (C(n, j)-C(n, j+1))*a(j).
a(n) = A000110(n) - A171859(n). - Emeric Deutsch, May 01 2010
G.f.: x*( 1 + (G(0)+1)*x/(1-x) ) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k+x-1) - x*(2*k+1)*(2*k+3)*(2*x*k+x-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+2*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 20 2012
G.f.: x*G(0)/(1-x^2) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-1) - x*(2*k+1)*(2*k+3)*(2*x*k-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 22 2012
G.f.: x*( G(0) - 1 )/(1-x) where G(k) = 1 + (1-x)/(1-x*k)/(1-x/(x+(1-x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 21 2013
G.f.: (G(0)-1)*x/(1-x^2) where G(k) = 1 + 1/(1-k*x)/(1-x/(x+1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Feb 06 2013
G.f.: x/(1-x)/(1-x*Q(0)), where Q(k) = 1 + x/(1 - x + x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
E.g.f. A(x) satisfies: A'(x) = A(x) + exp(exp(x)-1). - Geoffrey Critzer, Feb 04 2014
G.f.: (x/(1 - x)) * Sum_{i>=0} x^i / Product_{j=1..i} (1 - j*x). - Ilya Gutkovskiy, Jun 05 2017
a(n) ~ Bell(n) / (n/LambertW(n) - 1). - Vaclav Kotesovec, Jul 28 2021

A126390 a(n) = Sum_{i=0..n} 2^iB(i)binomial(n,i) where B(n) = Bell numbers A000110(n).

Original entry on oeis.org

1, 3, 13, 71, 457, 3355, 27509, 248127, 2434129, 25741939, 291397789, 3510328695, 44782460313, 602513988107, 8518757813637, 126179029108463, 1952609274344353, 31492811964616163, 528249539951292461, 9197240228562763687, 165923214676585626729

Offset: 0

N. J. A. Sloane, Aug 04 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Toufik Mansour and Mark Shattuck, A recurrence related to the Bell numbers, INTEGERS 11 (2011), #A67.

Cf. A000110, A000296, A005493, A124311, A126617, A284859, A285064.

with(combstruct):seq(count(([S, {N=Union(Z, S, P), S=Set(Union(Z, P), card>=0), P=Set(Union(Z, Z), card>=1)}, labeled], size=n)), n=0..20); # Zerinvary Lajos, Mar 18 2008

Table[ Sum[ 2^k Binomial[n, k] BellB[k], {k, 0, n}], {n, 0, 30}] (* Karol A. Penson and Olivier Gérard, Oct 22 2007 *)

x='x+O('x^66); Vec(serlaplace((exp(exp(2*x)-1+x)))) \\ Joerg Arndt, May 13 2013

E.g.f.: exp(exp(2*x)-1+x). - Vladeta Jovovic, Aug 04 2007
a(n) = e^(-1)* 2^n * Sum_{k>=0} (k + 1/2)^n / k!. This is a Dobinski-type formula. - Karol A. Penson and Olivier Gérard, Oct 22 2007
G.f.: 1/Q(0), where Q(k)= 1 - (2*k+3)*x - 4*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: 1/Q(0), where Q(k)= 1 - x - 2*x/(1 - 2*x*(2*k+1)/(1 - x - 2*x/(1 - 2*x*(2*k+2)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, May 13 2013
a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 2^k * a(n-k). - Ilya Gutkovskiy, Jun 21 2022
From Vaclav Kotesovec, Jun 22 2022: (Start)
a(n) ~ Bell(n) * (2 + LambertW(n)/n)^n.
a(n) ~ Bell(n) * 2^n * sqrt(n) * log(n)^(-1/2 + 1/(2*log(n)) - 1/(2*log(n)^2)) * exp(log(log(n))^2/(4*log(n)^2)). (End)
a(n) ~ 2^n * n^(n + 1/2) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 1/2)). - Vaclav Kotesovec, Jun 27 2022

A126617 a(n) = Sum_{i=0..n} (-2)^(n-i)B(i)binomial(n,i) where B(n) = Bell numbers A000110(n).

Original entry on oeis.org

1, -1, 2, -3, 7, -10, 31, -21, 204, 307, 2811, 12100, 74053, 432211, 2768858, 18473441, 129941283, 956187814, 7351696139, 58897405759, 490681196604, 4242903803727, 38014084430983, 352341755256348, 3373662303816313, 33326335433122711, 339232538387804530

Offset: 0

N. J. A. Sloane, Aug 04 2007

sign

a(n) is positive starting at n=8. - Karol A. Penson and Olivier Gérard, Oct 22 2007

Hankel transform is A000178. - Paul Barry, Apr 23 2009

			G.f.: 1 - 1*x + 2*x^2 - 3*x^3 + 7*x^4 - 10*x^5 + 31*x^6 - 21*x^7 + 204*x^8 + 307*x^9 + 2811*x^10 + 12100*x^11 + 74053*x^12 + 432211*x^13 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000110, A000296, A005493, A124311, A126390, A153732.

Table[ Sum[ (-2)^(n - k) Binomial[n, k] BellB[k], {k, 0, n}], {n, 0, 50}] (* Karol A. Penson and Olivier Gérard, Oct 22 2007 *)
With[{nn=30},CoefficientList[Series[Exp[Exp[x]-2x-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 19 2025 *)

E.g.f.: exp(exp(x)-2*x-1). - Vladeta Jovovic, Aug 04 2007
a(n) = e^(-1) * Sum_{k>=0} (k-2)^n / k!. This is a Dobinski-type formula. - Karol A. Penson and Olivier Gérard, Oct 22 2007
G.f.: 1/(1+x-x^2/(1-2x^2/(1-x-3x^2/(1-2x-4x^2/(1-3x-5x^2/(1-.... (continued fraction). - Paul Barry, Apr 23 2009
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>=1, a(n)=(-1)^(n)charpoly(A,2). - Milan Janjic, Jul 08 2010
G.f.: -1/U(0)  where U(k) = x*k - 1 - x - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 28 2012
G.f.: 1/G(0) where G(k) = 1 + 2*x/(1 + 1/(1 - 2*x*(k+1)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 23 2012
G.f.: G(0)/(1+3*x) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-2*x-1) - x*(2*k+1)*(2*k+3)*(2*x*k-2*x-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k-x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012
From Sergei N. Gladkovskii, Feb 13 2013: (Start)
Conjecture: if the e.g.f. is E(x)= exp( exp(x) -1 + p*x) then
g.f.: (x+1-p*x)/x/(G(0)-x) - 1/x where G(k) = 2*x + 1 - p*x - x*k + x*(x*k - x - 1 + p*x)/G(k+1); (continued fraction).
So, for this sequence (p=-2), g.f.: (3*x+1)/x/( G(0)-x ) - 1/x where G(k) = 4*x + 1 - x*k + x*(x*k - 3*x - 1)/G(k+1);
(End)
G.f.: 1/Q(0), where Q(k) = 1 + 2*x - x/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
a(0) = 1; a(n) = -2 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 30 2021
a(n) ~ n^(n-2) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n-2)). - Vaclav Kotesovec, Jun 27 2022

More terms from Karol A. Penson and Olivier Gérard, Oct 22 2007

A058692 a(n) = B(n) - 1, where B(n) = Bell numbers, A000110.

Original entry on oeis.org

1, 4, 14, 51, 202, 876, 4139, 21146, 115974, 678569, 4213596, 27644436, 190899321, 1382958544, 10480142146, 82864869803, 682076806158, 5832742205056, 51724158235371, 474869816156750, 4506715738447322, 44152005855084345

Offset: 2

N. J. A. Sloane, Dec 30 2000

nonn

			G.f. = x^2 + 4*x^3 + 14*x^4 + 51*x^5 + 202*x^6 + 876*x^7 + 4139*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 2..200
W. M. B. Dukes, Tables of matroids.
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
W. M. B. Dukes, On the number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
Index entries for sequences related to matroids

Column k=2 of both A058710 and A058711 (which are the same except for column k=0).

Cf. A000110.

[Bell(n)-1: n in [2..30]]; // Vincenzo Librandi, Mar 04 2014

A058692 := proc(n)
    combinat[bell](n)-1 ;
end proc:
seq(A058692(n),n=2..40) ; # R. J. Mathar, May 25 2025

Table[BellB[n, 1] - 1, {n, 2, 23}] (* Zerinvary Lajos, Jul 16 2009 *)

G.f.: Sum_{k > 1} x^k / ((1 - x) * (1 - x^2) * ... * (1 - x^k)). - Michael Somos, Feb 26 2014

E.g.f.: exp(exp(x) - 1) - exp(x). - Ilya Gutkovskiy, Feb 08 2020

A106436 Difference array of Bell numbers A000110 read by antidiagonals.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 2, 3, 5, 4, 5, 7, 10, 15, 11, 15, 20, 27, 37, 52, 41, 52, 67, 87, 114, 151, 203, 162, 203, 255, 322, 409, 523, 674, 877, 715, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 3425, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147

Offset: 0

Philippe Deléham, May 29 2005

Essentially Aitken's array A011971 with first column A000296.

Mirror image of A182930. - Alois P. Heinz, Jan 29 2019

			   1;
   0,  1;
   1,  1,  2;
   1,  2,  3,  5;
   4,  5,  7, 10, 15;
  11, 15, 20, 27, 37, 52;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Cf. A000110, A182930.
Diagonals give A005493, A011965-A011967, A191099, A000298, A011968-A011970.
T(2n,n) gives A020556.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
T:= proc(n, k) option remember; `if`(k=0, b(n),
      T(n+1, k-1)-T(n, k-1))
    end:
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Jan 29 2019

bb = Array[BellB, m = 12, 0];
dd[n_] := Differences[bb, n];
A = Array[dd, m, 0];
Table[A[[n-k+1, k+1]], {n, 0, m-1}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019 *)
a[0,0]:=1; a[n_,0]:=a[n-1,n-1]-a[n-1,0]; a[n_,k_]/;0Oliver Seipel, Nov 23 2024 *)

Double-exponential generating function: sum_{n, k} a(n-k, k) x^n/n! y^k/k! = exp(exp{x+y}-1-x). a(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,i-k)*Bell(i). - Vladeta Jovovic, Oct 14 2006

A124311 a(n) = Sum_{i=0..n} (-2)^ibinomial(n,i)B(i) where B(n) = Bell numbers A000110(n).

Original entry on oeis.org

1, -1, 5, -21, 121, -793, 5917, -49101, 447153, -4421105, 47062773, -535732805, 6484924585, -83079996041, 1121947980173, -15915567647101, 236442490569825, -3668776058118881, 59316847871113445, -997182232031471477, 17397298225094055897, -314449131128077197561

Offset: 0

N. J. A. Sloane, Aug 04 2007

sign

The sequence has strictly alternating signs. The variant Dobinski-type formula e^(-1)* (2)^n * Sum_{k >= 0} ( (k-1/2)^n / k! ) is strictly positive. - Karol A. Penson and Olivier Gérard, Oct 22 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000110, A000296, A005493, A126390, A126617.

A124311:= func< n | (&+[(-2)^k*Binomial(n,k)*Bell(k): k in [0..n]]) >;
[A124311(n): n in [0..30]]; // G. C. Greubel, Aug 25 2023

Table[ Sum[ (-2)^(k) Binomial[n, k] BellB[k], {k, 0, n}], {n, 0, 50}] (* Karol A. Penson and Olivier Gérard, Oct 22 2007 *)
With[{nn=30},CoefficientList[Series[Exp[Exp[-2x]-1+x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 04 2016 *)

def A124311_list(n):  # n>=1
    T = [0]*(n+1); R = [1]
    for m in (1..n-1):
        a,b,c = 1,0,0
        for k in range(m,-1,-1):
            r = a + 2*(k*(b+c)+c)
            if k < m : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u;
        R.append((-1)^m*sum(T))
    return R
A124311_list(22)  # Peter Luschny, Nov 02 2012

def A124311(n): return sum( (-2)^k*binomial(n,k)*bell_number(k) for k in range(n+1) )
[A124311(n) for n in range(31)] # G. C. Greubel, Aug 25 2023

E.g.f.: exp(exp(-2*x) - 1 + x). - Vladeta Jovovic, Aug 04 2007
G.f.: 1/U(0) where U(k)=  1 + x*(2*k+1) - 4*x^2*(k+1)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
a(n) ~ (-2)^n * n^(n - 1/2) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n - 1/2)). - Vaclav Kotesovec, Jun 26 2022
a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * (-2)^k * a(n-k). - Ilya Gutkovskiy, Nov 29 2023

cp's OEIS Frontend

A054767 Period of the sequence of Bell numbers A000110 (mod n).

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A146093 Bell numbers (A000110) read mod 3.

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Magma

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A146122 Bell numbers (A000110) read mod 32.

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Magma

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A303601 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + Bell(k) + Bell(m) with 0 <= a <= b and 0 < k <= m, where Bell(k) denotes the k-th Bell number A000110(k).

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A005001 a(n) = Sum_{k=0..n-1} Bell(k), where the Bell numbers Bell(k) are given in A000110.

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A126390 a(n) = Sum_{i=0..n} 2^i*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n).

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A126617 a(n) = Sum_{i=0..n} (-2)^(n-i)*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n).

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A058692 a(n) = B(n) - 1, where B(n) = Bell numbers, A000110.

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A303601 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + Bell(k) + Bell(m) with 0 <= a <= b and 0 < k <= m, where Bell(k) denotes the k-th Bell number A000110(k).

A126390 a(n) = Sum_{i=0..n} 2^iB(i)binomial(n,i) where B(n) = Bell numbers A000110(n).

A126617 a(n) = Sum_{i=0..n} (-2)^(n-i)B(i)binomial(n,i) where B(n) = Bell numbers A000110(n).

A124311 a(n) = Sum_{i=0..n} (-2)^ibinomial(n,i)B(i) where B(n) = Bell numbers A000110(n).