A011970 -id:A011970 - cp's OEIS frontend

A106436 Difference array of Bell numbers A000110 read by antidiagonals.

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

A011968 Apply (1+Shift) to Bell numbers.

Original entry on oeis.org

1, 2, 3, 7, 20, 67, 255, 1080, 5017, 25287, 137122, 794545, 4892167, 31858034, 218543759, 1573857867, 11863100692, 93345011951, 764941675963, 6514819011216, 57556900440429, 526593974392123, 4981585554604074, 48658721593531669, 490110875149889635

Offset: 0

N. J. A. Sloane

nonn

Number of set partitions of n+2 with at least one singleton and the smallest element in any singleton is exactly n. The maximum number of singletons is therefore 3. Alternatively, number of set partitions of n+2 with at least one singleton and the largest element in any singleton is exactly 3 (or n+2 if n+2 < 3). For example, a(3)=7 counts the following set partitions of [5]: {1245, 3}, {12, 3, 45}, {124, 3, 5}, {15, 24, 3}, {125, 3, 4}, {14, 25, 3}, {12, 3, 4, 5}. - Olivier Gérard, Oct 29 2007

Let V(N)={v(1),v(2),...,v(N)} denote an ordered set of increasing positive integers containing a pair of adjacent elements that differ by at least 2, that is, v(i),v(i+1) with v(i+1)-v(i) > 1. Then for n > 0, a(n) is the number of partitions of V(n+1) into blocks of nonconsecutive integers. - Augustine O. Munagi, Jul 17 2008

			a(3)=7 because the set {1,3,4,5} has 7 different partitions into blocks of nonconsecutive integers: 14/35, 135/4, 1/35/4, 13/4/5, 14/3/5, 15/3/4, 1/3/4/5.

Olivier Gérard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011

Chai Wah Wu, Table of n, a(n) for n = 0..500 n = 0..200 from Vincenzo Librandi.
Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841.
Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841. [Annotated scanned copy]
Augustine O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 1298--1308.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Cf. A000110, A000296, A011969, A011970.
A diagonal of A011971 and A106436. - N. J. A. Sloane, Jul 31 2012
Cf. A000569, A240936, A321750.

with(combinat): seq(`if`(n>0,bell(n)+bell(n-1),1),n=0..21); # Augustine O. Munagi, Jul 17 2008

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011968_list, blist, b = [1,2], [1], 1
for _ in range(10**2):
    blist = list(accumulate([b]+blist))
    A011968_list.append(b+blist[-1])
    b = blist[-1] # Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

For n >= 1, a(n+1) = exp(-1)*Sum_{k>=0} ((k+1)/k!)*k^n. - Benoit Cloitre, Mar 09 2008
For n >= 1, a(n) = Bell(n) + Bell(n-1). - Augustine O. Munagi, Jul 17 2008
G.f.: G(0) 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) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012
G.f.: 1 + x*E(0) where E(k) = 1 + 1/(1-x*k-x)/(1-x/(x+1/E(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 20 2013
G.f.: 1 + Sum_{k>=0} ( 1+1/(1-x-x*k) )*x^(k+1)/Product_{i=0..k} (1-x*i). - Sergei N. Gladkovskii, Jan 20 2013
a(n) ~ Bell(n) * (1 + LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021

A011969 Apply (1+Shift)^2 to Bell numbers.

Original entry on oeis.org

1, 3, 5, 10, 27, 87, 322, 1335, 6097, 30304, 162409, 931667, 5686712, 36750201, 250401793, 1792401626, 13436958559, 105208112643, 858286687914, 7279760687179, 64071719451645, 584150874832552, 5508179528996197

Offset: 0

N. J. A. Sloane

nonn

Starting with n=2 (a(2)=5), number of set partitions of n+2 with at least one singleton and the smallest element in any singleton is exactly n-1. The maximum number of singletons is therefore 4. Alternatively, starting with n=2, number of set partitions of n+2 with at least one singleton and the largest element in any singleton is exactly 4. E.g. a(3)=10 counts the following set partitions of [5]: {1345, 2}, {13, 2, 45}, {145, 2, 3}, {134, 2, 5}, {15, 2, 34}, {135, 2, 4}, {14, 2, 35}, {13, 2, 4, 5}, {14, 2, 3, 5}, {15, 2, 3, 4}. - Olivier Gérard, Oct 29 2007

Let V(N)={v(1),v(2),...,v(N)} denote an ordered set of increasing positive integers containing 2 pairs of adjacent elements that differ by at least 2, that is, v(i),v(i+1) with v(i+1)-v(i)>1. Then for n>1, a(n) is the number of partitions of V(n+1) into blocks of nonconsecutive integers. - Augustine O. Munagi, Jul 17 2008

			a(3)=10 because the set {1,3,5,6} has 10 different partitions into blocks of nonconsecutive integers: 15/36, 16/35, 135/6, 136/5, 1/35/6, 1/36/5, 13/5/6, 15/3/6, 16/3/5, 1/3/5/6.

Olivier Gérard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011

Chai Wah Wu, Table of n, a(n) for n = 0..500 n = 0..200 from Vincenzo Librandi.
Martin Cohn, Shimon Even, Karl Menger, Jr., and Philip K. Hooper, On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841.
Martin Cohn, Shimon Even, Karl Menger, Jr., and Philip K. Hooper, On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841. [Annotated scanned copy]
Augustine O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 1298--1308.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Cf. A000110, A011968, A011970.

A diagonal of A011971 and A106436. - N. J. A. Sloane, Jul 31 2012

with(combinat): 1,3,seq(`if`(n>1,bell(n)+2*bell(n-1)+bell(n-2),NULL),n=2..22); # Augustine O. Munagi, Jul 17 2008

Join[{1,3},#[[1]]+2#[[2]]+#[[3]]&/@Partition[BellB[Range[0,30]],3,1]] (* Harvey P. Dale, May 05 2023 *)

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011969_list, blist, b, b2 = [1,3], [1], 1, 1
for _ in range(10**2):
    blist = list(accumulate([b]+blist))
    A011969_list.append(2*b+b2+blist[-1])
    b2, b = b, blist[-1]
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

For n >= 1, a(n+2) = exp(-1)*Sum_{k>=0} (k+1)^2/k!*k^n. - Benoit Cloitre, Mar 09 2008
If n>1, then a(n) = Bell(n) + 2*Bell(n-1) + Bell(n-2). - Augustine O. Munagi, Jul 17 2008
G.f.: -(1+2*x)*(1+x)^2*Sum_{k>=0} x^(2*k)*(4*x*k^2-2*k-2*x-1)/((2*k+1)*(2*x*k-1))*A(k)/B(k) where A(k) = Product_{p=0..k} (2*p+1), B(k) = Product_{p=0..k} (2*p-1)*(2*x*p-x-1)*(2*x*p-2*x-1). - Sergei N. Gladkovskii, Jan 03 2013 [corrected by Jason Yuen, Apr 03 2025]
G.f.: G(0)*(1+x) 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, Jan 03 2013
a(n) ~ Bell(n) * (1 + 2*LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021

cp's OEIS Frontend

A106436 Difference array of Bell numbers A000110 read by antidiagonals.

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A011968 Apply (1+Shift) to Bell numbers.

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A011969 Apply (1+Shift)^2 to Bell numbers.

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Programs

Maple

Mathematica

Python

Formula