A000296 -id:A000296 - cp's OEIS frontend

A006505 Number of partitions of an n-set into boxes of size >2.

1, 0, 0, 1, 1, 1, 11, 36, 92, 491, 2557, 11353, 60105, 362506, 2169246, 13580815, 91927435, 650078097, 4762023647, 36508923530, 292117087090, 2424048335917, 20847410586719, 185754044235873, 1711253808769653, 16272637428430152

Offset: 0

N. J. A. Sloane

J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, April 4-7, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..579 (terms 0..250 from Alois P. Heinz)
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 102
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
J. Riordan, Cached copy of paper [With permission]

Column k=2 of A293024.
Cf. A000110, A000296, A057814, A057837, A059022.
Cf. A293038.

Copy ZL := [ B,{B=Set(Set(Z, card>=3))}, labeled ]: [seq(combstruct[count](ZL, size=n), n=0..25)]; # Zerinvary Lajos, Mar 13 2007
G:={P=Set(Set(Atom,card>=3))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..25); # Zerinvary Lajos, Dec 16 2007
g:=proc(n) option remember; if n=0 then RETURN(1); fi; if n<=2 then RETURN(0); fi; if n<=5 then RETURN(x); fi; expand(x*add(binomial(n-1,i)*g(i),i=0..n-3)); end; [seq(subs(x=1,g(n)),n=0..60)]; # N. J. A. Sloane, Jul 20 2011

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ Exp @ x - 1 - x - x^2 / 2], {x, 0, n}]] (* Michael Somos, Jul 20 2011 *)
a[0] = 1; a[n_] := n!*Sum[Sum[k!*(-1)^(m-k)*Binomial[m, k]*Sum[StirlingS2[i+k, k]* Binomial[m-k, n-m-i]*2^(-n+m+i)/(i+k)!, {i, 0, n-m}], {k, 0, m}]/m!, {m, 1, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)
Table[Sum[(-1)^j * Binomial[n, j] * BellB[n-j] * 2^((j-1)/2) * HypergeometricU[(1 - j)/2, 3/2, 1/2], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Feb 09 2020 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) - 1 - x - x^2 / 2), n))} /* Michael Somos, Jul 20 2011 */

E.g.f.: exp ( exp x - 1 - x - (1/2)*x^2 ).
a(n) = Sum_{k=1..[n/3]} A059022(n,k), n>=3. - R. J. Mathar, Nov 08 2008
a(n) = n! * sum(m=1..n, sum(k=0..m, k!*(-1)^(m-k) *binomial(m,k) *sum(i=0..n-m, stirling2(i+k,k) *binomial(m-k,n-m-i) *2^(-n+m+i)/ (i+k)!))/m!); a(0)=1. - Vladimir Kruchinin, Feb 01 2011
Define polynomials g_n by g_0=1, g_1=g_2=0, g_3=g_4=g_5=x; g(n) = x*Sum_{i=0..n-3} binomial(n-1,i)*g_i; then a(n) = g_n(1). [Riordan]
a(0) = 1; a(n) = Sum_{k=0..n-3} binomial(n-1,k+2) * a(n-k-3). - Seiichi Manyama, Sep 22 2023

More terms from Christian G. Bower, Nov 09 2000

Edited by N. J. A. Sloane, Jul 20 2011

A050370 Number of ways to factor n into composite factors.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 3, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 3, 0, 1, 1, 4, 1, 1, 0, 1, 1, 1, 0, 4, 0, 1, 1, 1, 1, 1, 0, 3, 2, 1, 0, 3, 1, 1, 1, 2, 0, 3, 1, 1, 1, 1, 1, 5, 0, 1, 1, 3, 0, 1

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A001055, A002808, A050371, A050372, A050373, A050374, A050375.

a(p^k)=A002865. a(A002110)=A000296.

with(numtheory):
g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
         d=divisors(n) minus {1, n}))
    end:
a:= proc(n) a(n):= add(mobius(n/d)*g(d$2), d=divisors(n)) end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 16 2014

g[n_, k_] := g[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, g[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]]; a[n_] := Sum[ MoebiusMu[n/d]*g[d, d], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 23 2017, after Alois P. Heinz *)

from sympy.core.cache import cacheit
from sympy import mobius, divisors, isprime
@cacheit
def g(n, k): return (0 if n>k else 1) + (0 if isprime(n) else sum((0 if d>k else g(n//d, d)) for d in divisors(n)[1:-1]))
def a(n): return sum(mobius(n//d)*g(d, d) for d in divisors(n))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 19 2017, after Maple code

Dirichlet g.f.: Product_{n is composite}(1/(1-1/n^s)).

Moebius transform of A001055. - Vladeta Jovovic, Mar 17 2004

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

A105794 Inverse of a generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -1, 1, 1, -1, 1, -1, 1, 0, 1, 1, -1, 1, 2, 1, -1, 1, 0, 5, 5, 1, 1, -1, 1, 10, 20, 9, 1, -1, 1, 0, 21, 70, 56, 14, 1, 1, -1, 1, 42, 231, 294, 126, 20, 1, -1, 1, 0, 85, 735, 1407, 924, 246, 27, 1, 1, -1, 1, 170, 2290, 6363, 6027, 2400, 435, 35, 1

Offset: 0

Paul Barry, Apr 20 2005

Inverse of number triangle A105793. Row sums are the generalized Bell numbers A000296.
T(n,k)*(-1)^(n-k) gives the inverse Sheffer matrix of A094645. In the umbral notation (cf. Roman, p. 17, quoted in A094645) this is Sheffer for (1-t,-log(1-t)). - Wolfdieter Lang, Jun 20 2011
From Peter Bala, Jul 10 2013: (Start)
For a graph G and a positive integer k, the graphical Stirling number S(G;k) is the number of partitions of the vertex set of G into k nonempty independent sets (an independent set in G is a subset of the vertices, no two elements of which are adjacent). Omitting the first two rows and columns of this triangle produces the triangle of graphical Stirling numbers of cycles on n vertices (interpreting a 2-cycle as a single edge). In comparison, the classical Stirling numbers of the second kind, A008277, are the graphical Stirling numbers of path graphs on n vertices. See Galvin and Than. See also A227341.
This is the triangle of connection constants for expressing the polynomial sequence (x-1)^n in terms of the falling factorial polynomials, that is, (x-1)^n = Sum_{k = 0..n} T(n,k)*x_(k), where x_(k) = x*(x-1)*...*(x-k+1) denotes the falling factorial.
The row polynomials are a particular case of the Actuarial polynomials - see Roman 4.3.4.  (End)

			The triangle starts with
  n=0:  1;
  n=1: -1,  1;
  n=2:  1, -1, 1;
  n=3: -1,  1, 0, 1;
  n=4:  1, -1, 1, 2, 1;
  n=5: -1,  1, 0, 5, 5, 1;
  ... - _Wolfdieter Lang_, Jun 20 2011

S. Roman, The umbral calculus, Pure and Applied Mathematics 111, Academic Press Inc., New York, 1984. Reprinted by Dover in 2005.

Robert Israel, Table of n, a(n) for n = 0..9869
Peter Bala, Notes on A105794
Peter Bala, A 3 parameter family of generalized Stirling numbers
B. Duncan and R. Peele, Bell and Stirling Numbers for Graphs, Journal of Integer Sequences 12 (2009), article 09.7.1.
D. Galvin and D. T. Thanh, Stirling numbers of forests and cycles, Electr. J. Comb. Vol. 20(1): P73 (2013).
Sophie Morier-Genoud, Counting Coxeter's friezes over a finite field via moduli spaces, arXiv:1907.12790 [math.CO], 2019.

Cf. A000296 (row sums), A105793 (matrix inverse), A227341.

Cf. A007318, A008277, A048993.

B:= Matrix(12,12,shape=triangular[lower],(n,k) -> combinat:-stirling1(n-1,k-1)+(n-1)*combinat:-stirling1(n-2,k-1)):
A:= B^(-1):
seq(seq(A[i,j],j=1..i),i=1..12); # Robert Israel, Jan 19 2015
T := (n, k) -> add((-1)^(n - i)*binomial(n, i)*Stirling2(i, k), i=0..n):
seq(seq(T(n, k), k=0..n), n=0..9);  # Peter Luschny, Feb 15 2025

Table[Sum[(-1)^(n - i)*Binomial[n, i] StirlingS2[i, k], {i, 0, n}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 14 2019 *)

Term k in row n is given by {(-1)^(k+n) * (Sum_{j=0..k} (-1)^j * binomial(k,j) * (1-j)^n) / k! }; i.e., a finite difference. - Tom Copeland, Jun 05 2006
O.g.f. for row n = n-th finite difference of the Touchard (Bell) polynomials, T(x,j) and so the e.g.f. for these finite differences and therefore the sequence = exp{x*[exp(t)-1]-t} = exp{t*[T(x,.)-1]} umbrally. - Tom Copeland, Jun 05 2006
The e.g.f. A(x,t) = exp(x*(exp(t)-1)-t) satisfies the partial differential equation x*dA/dx - dA/dt = (1-x)*A.
Recurrence relation: T(n+1,k) = T(n,k-1) + (k-1)*T(n,k).
Let f(x) = ((x-1)/x)*exp(x). For n >= 1, the n-th row polynomial R(n,x) = x*exp(-x)*(x*d/dx)^(n-1)(f(x)) and satisfies the recurrence equation R(n+1,x) = (x-1)*R(n,x) + x*R'(n,x). - Peter Bala, Oct 28 2011
Let f(x) = exp(exp(x)-x). Then R(n,exp(x)) = 1/f(x)*(d/dx)^n(f(x)). Similar formulas hold for A008277, A039755, A111577, A143494 and A154537. - Peter Bala, Mar 01 2012
From Peter Bala, Jul 10 2013: (Start)
T(n,k) = Sum_{i = 0..n-1} (-1)^i*Stirling2(n-1-i,k-1), for n >= 1, k >= 1.
The k-th column o.g.f. is (1/(1+x))*x^k/((1-x)*(1-2*x)*...*(1-(k-1)*x)) (the empty product occurring in the denominator when k = 0 and k = 1 is taken equal to 1).
Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k >= 0} (k-1)^n*x^k/k!.
Sum_{k = 0..n} binomial(n,k)*R(k,x) = n-th Bell polynomial (n-th row polynomial of A048993). (End)
From Peter Bala, Jan 13 2014: (Start)
T(n,k) = Sum_{i = 0..n} (-1)^(n-i)*binomial(n,i)*Stirling2(i,k).
T(n,k) = Sum_{i = 0..n} (-2)^(n-i)*binomial(n,i)*Stirling2(i+1,k+1).
Matrix product P^(-1) * S = P^(-2) * S1, where P = A007318, S = A048993 and S1 = A008277. (End)
From Werner Schulte, Feb 15 2025: (Start)
Conjecture 1: Sum_{k=0..n} (-1)^(n-k) * T(n, k) * (k+m)! / m! = (m+2)^n for m >= 0.
Conjecture 2: (-1)^n - B(n) = Sum_{k=1..n} (-1)^k * T(n, k) * (k-1)! / (k+1) where B(n) = B(n, 0) is n-th Bernoulli number. (End)

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

A324166 Number of totally crossing set partitions of {1,...,n}.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 18, 57, 207, 842, 3673, 17062, 84897

Offset: 0

Gus Wiseman, Feb 17 2019

A set partition is totally crossing if every pair of distinct blocks is of the form {{...x...y...}, {...z...t...}} for some x < z < y < t or z < x < t < y.

			The a(6) = 18 totally crossing set partitions:
  {{1,2,3,4,5,6}}
  {{1,4,6},{2,3,5}}
  {{1,4,5},{2,3,6}}
  {{1,3,6},{2,4,5}}
  {{1,3,5},{2,4,6}}
  {{1,3,4},{2,5,6}}
  {{1,2,5},{3,4,6}}
  {{1,2,4},{3,5,6}}
  {{4,6},{1,2,3,5}}
  {{3,6},{1,2,4,5}}
  {{3,5},{1,2,4,6}}
  {{2,6},{1,3,4,5}}
  {{2,5},{1,3,4,6}}
  {{2,4},{1,3,5,6}}
  {{1,5},{2,3,4,6}}
  {{1,4},{2,3,5,6}}
  {{1,3},{2,4,5,6}}
  {{1,4},{2,5},{3,6}}

Cf. A000108 (non-crossing partitions), A000110, A000296, A002662, A016098 (crossing partitions), A054726, A099947 (topologically connected partitions), A305854, A306006, A306418, A306438, A319752.

Cf. A324170, A324171, A324172, A324173.

nn=6;
nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

A057814 Number of partitions of an n-set into blocks of size > 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 127, 463, 1255, 3004, 6722, 140570, 1039260, 5371627, 23202077, 90048525, 814737785, 7967774337, 62895570839, 417560407223, 2455461090505, 18440499041402, 179627278800426, 1770970802250146

Offset: 0

Steven C. Fairgrieve (fsteven(AT)math.wvu.edu), Nov 06 2000

Seiichi Manyama, Table of n, a(n) for n = 0..589 (terms 0..300 from Alois P. Heinz)
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.

Column k=4 of A293024.
Row sums of A059024.
Cf. A000110, A000296, A006505, A057837.
Cf. A293040.

G:={P=Set(Set(Atom,card>=5))}:combstruct[gfsolve](G,labeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..27); # Zerinvary Lajos, Dec 16 2007

max = 27; CoefficientList[ Series[ Exp[ Exp[x] - Normal[ Series[ Exp[x], {x, 0, 4}]]], {x, 0, max}], x]*Range[0, max]!(* Jean-François Alcover, Apr 25 2012, from e.g.f. *)

E.g.f.: exp(exp(x)-1-x-x^2/2-x^3/6-x^4/24).

a(0) = 1; a(n) = Sum_{k=5..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020

A057837 Number of partitions of a set of n elements where the partitions are of size > 3.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 36, 127, 337, 793, 7525, 48764, 238954, 997790, 6401435, 49107697, 345482807, 2150694855, 14656830110, 116678887407, 978172378669, 7886661080873, 63905475745765, 553437891603452, 5122279358273976, 48331088541366296, 458771027309344261

Offset: 0

Steven C. Fairgrieve (fsteven(AT)math.wvu.edu), Nov 06 2000

Seiichi Manyama, Table of n, a(n) for n = 0..583 (terms 0..250 from Alois P. Heinz)
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1.

Column k=3 of A293024.
Row sums of A059023.
Cf. A000110, A000296, A006505, A057814.
Cf. A293039.

G:={P=Set(Set(Atom,card>=4))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..26); # Zerinvary Lajos, Dec 16 2007

With[{nn=30},CoefficientList[Series[Exp[Exp[x]-1-x-x^2/2-x^3/6],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Jun 28 2012 *)

E.g.f.: exp(exp(x)-1-x-x^2/2-x^3/6).

a(0) = 1; a(n) = Sum_{k=4..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020

Corrected and extended by Christian G. Bower and James Sellers, Nov 09 2000

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

A006505 Number of partitions of an n-set into boxes of size >2.

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A050370 Number of ways to factor n into composite factors.

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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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A105794 Inverse of a generalized Stirling number triangle of first kind.

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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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A324166 Number of totally crossing set partitions of {1,...,n}.

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A057814 Number of partitions of an n-set into blocks of size > 4.

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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).