A005493 -id:A005493 - cp's OEIS frontend

A003436 Number of inequivalent labeled Hamiltonian circuits on n-octahedron. Interlacing chords joining 2n points on circle.

1, 0, 1, 4, 31, 293, 3326, 44189, 673471, 11588884, 222304897, 4704612119, 108897613826, 2737023412199, 74236203425281, 2161288643251828, 67228358271588991, 2225173863019549229, 78087247031912850686, 2896042595237791161749, 113184512236563589997407

Offset: 0

N. J. A. Sloane

Also called the relaxed ménage problem (cf. A000179).
a(n) can be seen as a subset of the unordered pairings of the first 2n integers (A001147) with forbidden pairs (1,2n) and (i,i+1) for all i in [1,2n-1] (all adjacent integers modulo 2n). The linear version of this constraint is A000806. - Olivier Gérard, Feb 08 2011
Number of perfect matchings in the complement of C_{2n} where C_{2n} is the cycle graph on 2n vertices. - Andrew Howroyd, Mar 15 2016
Also the number of 2-uniform set partitions of {1...2n} containing no two cyclically successive vertices in the same block. - Gus Wiseman, Feb 27 2019

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200
F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994
Kenneth P. Bogart and Peter G. Doyle, Nonsexist solution of the menage problem, Amer. Math. Monthly 93:7 (1986), 514-519.
Robert Cori and G. Hetyei, Counting partitions of a fixed genus, arXiv preprint arXiv:1710.09992 [math.CO], 2017.
M. Hazewinkel and V. V. Kalashnikov, Counting Interlacing Pairs on the Circle, CWI report AM-R9508 (1995)
Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017.
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv preprint arXiv:1601.05073 [math.CO], 2016.
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, The Electronic Journal of Combinatorics, 24(3) (2017), #P3.43.
D. Singmaster, Hamiltonian circuits on the n-dimensional octahedron, J. Combinatorial Theory Ser. B 19 (1975), no. 1, 1-4.
Gus Wiseman, The a(5) = 293 interlacing chord diagrams.

Cf. A003435, A129348. A003437 gives unlabeled case.
First differences of A000806.
Column k=2 of A324428.
Cf. A000179, A000296, A000699, A001147, A005493, A170941, A190823, A278990, A306386, A306419, A322402, A324011, A324172, A324173.

A003436 := proc(n) local k;
      if n = 0 then 1
    elif n = 1 then 0
    else add( (-1)^k*binomial(n,k)*2*n/(2*n-k)*2^k*(2*n-k)!/2^n/n!,k=0..n) ;
    end if;
end proc: # R. J. Mathar, Dec 11 2013
A003436 := n-> `if`(n<2, 1-n, (-1)^n*2*hypergeom([n, -n], [], 1/2)):
seq(simplify(A003436(n)), n=0..18); # Peter Luschny, Nov 10 2016

a[n_] := (2*n-1)!! * Hypergeometric1F1[-n, 1-2*n, -2]; a[1] = 0;
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Apr 05 2013 *)
twounifll[{}]:={{}};twounifll[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@twounifll[Complement[set,s]]]/@Table[{i,j},{j,If[i==1,Select[set,2<#i+1&]]}];
Table[Length[twounifll[Range[n]]],{n,0,14,2}] (* Gus Wiseman, Feb 27 2019 *)

a(n) = A003435(n)/(n!*2^n).
a(n) = 2*n*a(n-1)-2*(n-3)*a(n-2)-a(n-3) for n>4. [Corrected by Vasu Tewari, Apr 11 2010, and by R. J. Mathar, Oct 02 2013]
G.f.: x + ((1-x)/(1+x)) * Sum_{n>=0} A001147(n)*(x/(1+x)^2)^n. - Vladeta Jovovic, Jun 27 2007
a(n) ~ 2^(n+1/2)*n^n/exp(n+1). - Vaclav Kotesovec, Aug 13 2013
a(n) = (-1)^n*2*hypergeom([n, -n], [], 1/2) for n >= 2. - Peter Luschny, Nov 10 2016

a(0)=1 prepended by Gus Wiseman, Feb 27 2019

A278990 Number of loopless linear chord diagrams with n chords.

Original entry on oeis.org

1, 0, 1, 5, 36, 329, 3655, 47844, 721315, 12310199, 234615096, 4939227215, 113836841041, 2850860253240, 77087063678521, 2238375706930349, 69466733978519340, 2294640596998068569, 80381887628910919255, 2976424482866702081004, 116160936719430292078411

Offset: 0

N. J. A. Sloane, Dec 07 2016

nonn

See the signed version of these numbers, A000806, for much more information about these numbers.
From Gus Wiseman, Feb 27 2019: (Start)
Also the number of 2-uniform set partitions of {1..2n} containing no two successive vertices in the same block. For example, the a(3) = 5 set partitions are:
  {{1,3},{2,5},{4,6}}
  {{1,4},{2,5},{3,6}}
  {{1,4},{2,6},{3,5}}
  {{1,5},{2,4},{3,6}}
  {{1,6},{2,4},{3,5}}
(End)
From Gus Wiseman, Jul 05 2020: (Start)
Also the number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal and where the first i appears before the first j for i < j. For example, the a(3) = 5 permutations are the following.
  (1,2,3,1,2,3)
  (1,2,3,1,3,2)
  (1,2,3,2,1,3)
  (1,2,3,2,3,1)
  (1,2,1,3,2,3)
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..404 (terms 0..200 from Gheorghe Coserea)
Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2019.
H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.
E. Krasko, I. Labutin, and A. Omelchenko, Enumeration of labelled and unlabelled Hamiltonian Cycles in complete k-partite graphs, arXiv:1709.03218 [math.CO], 2017, Table 1.
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv:1601.05073 [math.CO], 2016.
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, The Electronic Journal of Combinatorics, 24(3) (2017), #P3.43.
Gus Wiseman, The a(4) = 36 loopless linear chord diagrams.
Donovan Young, Counting Bubbles in Linear Chord Diagrams, arXiv:2311.01569 [math.CO], 2023.
Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024.

Column k=0 of A079267.
Column k=2 of A293157.
Row n=2 of A322013.
Cf. A000110, A000699 (topologically connected 2-uniform), A000806, A001147 (2-uniform), A003436 (cyclical version), A005493, A170941, A190823 (distance 3+ version), A322402, A324011, A324172.
Anti-run compositions are A003242.
Separable partitions are A325534.
Cf. A007716, A292884, A333489.
Other sequences involving the multiset {1,1,2,2,...,n,n}: A001147, A007717, A020555, A094574, A316972.

[n le 2 select 2-n else (2*n-3)*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 26 2023

RecurrenceTable[{a[n]== (2n-1)a[n-1] +a[n-2], a[0]==1, a[1]==0}, a, {n,0,20}] (* Vaclav Kotesovec, Sep 15 2017 *)
FullSimplify[Table[-I*(BesselI[1/2+n,-1] BesselK[3/2,1] - BesselI[3/2,-1] BesselK[1/2+ n,1]), {n,0,20}]] (* Vaclav Kotesovec, Sep 15 2017 *)
Table[(2 n-1)!! Hypergeometric1F1[-n,-2 n,-2], {n,0,20}] (* Eric W. Weisstein, Nov 14 2018 *)
Table[Sqrt[2/Pi]/E ((-1)^n Pi BesselI[1/2+n,1] +BesselK[1/2+n,1]), {n,0,20}] // FunctionExpand // FullSimplify (* Eric W. Weisstein, Nov 14 2018 *)
twouniflin[{}]:={{}};twouniflin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@twouniflin[Complement[set,s]]]/@Table[{i,j},{j,Select[set,#>i+1&]}];
Table[Length[twouniflin[Range[n]]],{n,0,14,2}] (* Gus Wiseman, Feb 27 2019 *)

seq(N) = {
  my(a = vector(N)); a[1] = 0; a[2] = 1;
  for (n = 3, N, a[n] = (2*n-1)*a[n-1] + a[n-2]);
  concat(1, a);
};
seq(20) \\ Gheorghe Coserea, Dec 09 2016

def A278990_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-1+sqrt(1-2*x))/sqrt(1-2*x) ).egf_to_ogf().list()
A278990_list(30) # G. C. Greubel, Sep 26 2023

From Gheorghe Coserea, Dec 09 2016: (Start)
D-finite with recurrence a(n) = (2*n-1)*a(n-1) + a(n-2), with a(0) = 1, a(1) = 0.
E.g.f. y satisfies: 0 = (1-2*x)*y'' - 3*y' - y.
a(n) - a(n-1) = A003436(n) for all n >= 2. (End)
From Vaclav Kotesovec, Sep 15 2017: (Start)
a(n) = sqrt(2)*exp(-1)*(BesselK(1/2 + n, 1)/sqrt(Pi) - i*sqrt(Pi)*BesselI(1/2 + n, -1)), where i is the imaginary unit.
a(n) ~ 2^(n+1/2) * n^n  / exp(n+1). (End)
a(n) = A114938(n)/n! - Gus Wiseman, Jul 05 2020 (from Alexander Burstein's formula at A114938).
From G. C. Greubel, Sep 26 2023: (Start)
a(n) = (-1)^n * (i/e)*Sqrt(2/Pi) * BesselK(n + 1/2, -1).
G.f.: sqrt(Pi/(2*x)) * exp(-(1+x)^2/(2*x)) * Erfi((1+x)/sqrt(2*x)).
E.g.f.: exp(-1 + sqrt(1-2*x))/sqrt(1-2*x). (End)

a(0)=1 prepended by Gheorghe Coserea, Dec 09 2016

A005494 3-Bell numbers: E.g.f.: exp(3*z + exp(z) - 1).

Original entry on oeis.org

1, 4, 17, 77, 372, 1915, 10481, 60814, 372939, 2409837, 16360786, 116393205, 865549453, 6713065156, 54190360453, 454442481041, 3952241526188, 35590085232519, 331362825860749, 3185554606447814, 31581598272055879, 322516283206446897, 3389017736055752178

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

For further information, references, programs, etc. for r-Bell numbers see A005493. - N. J. A. Sloane, Nov 27 2013
From expansion of falling factorials (binomial transform of A005493).
Row sums of Sheffer triangle (exp(3*x), exp(x)-1). - Wolfdieter Lang, Sep 29 2011

			G.f. = 1 + 4*x + 17*x^2 + 77*x^3 + 372*x^4 + 1915*x^5 + 10481*x^6 + 60814*x^7 + ...

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
R. Jakimczuk, Successive Derivatives and Integer Sequences, J. Int. Seq. 14 (2011) # 11.7.3.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Toufik Mansour and Mark Shattuck, A recurrence related to the Bell numbers, INTEGERS 11 (2011), #A67.
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1.
J. Riordan, Letter, Oct 31 1977
N. J. A. Sloane, Transforms
Earl Glen Whitehead Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.

Cf. A000110, A005493, A045379, A108087, A124323, A196834.

A row or column of the array A108087.

A005494:= func< n | (&+[Binomial(n,j)*3^(n-j)*Bell(j): j in [0..n]]) >;
[A005494(n): n in [0..30]]; // G. C. Greubel, Dec 01 2022

seq(add(3^(n-i)*combinat:-bell(i)*binomial(n,i),i=0..n), n=0..50); # Robert Israel, Dec 16 2014
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0,
      m^2, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n+1, 0)-b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 03 2025

Range[0, 40]! CoefficientList[Series[Exp[3 x + Exp[x] - 1], {x, 0, 40}], x] (* Vincenzo Librandi, Mar 04 2014 *)

def A005494(n): return sum( 3^(n-j)*bell_number(j)*binomial(n,j) for j in range(n+1))
[A005494(n) for n in range(31)] # G. C. Greubel, Dec 01 2022

a(n) = Sum_{i=0..n} 3^(n-i)*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n). - Fred Lunnon, Aug 04 2007
a(n) = exp(-1)*Sum_{k>=0} ((k+3)^n)/k!. - Gerald McGarvey, Jun 03 2004. May be rewritten as a(n) = Sum_{k>=3} (k^n*(k-1)*(k-2)/k!)/exp(1), which is a Dobinski-type relation for this sequence. - Karol A. Penson, Aug 18 2006
Define f_1(x), f_2(x), ... such that f_1(x) = x^2*e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n-1) = e^(-1)*f_n(1). - Milan Janjic, May 30 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 >= 1, a(n) = (-1)^(n)charpoly(A,-3). - Milan Janjic, Jul 08 2010
a(n) = Sum_{k=3..n+3} A143495(n+3,k), n >= 0. - Wolfdieter Lang, Sep 29 2011
G.f.: 1/U(0) where U(k)= 1 - x*(k+4) - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: Sum_{k>0} x^(k-1) / ((1 - 3*x) * (1 - 4*x) * ... * (1 - (k+2)*x)). - Michael Somos, Feb 26 2014
G.f.: Sum_{k>0} k * x^(k-1) / ((1 - 2*x) * (1 - 3*x) * ... * (1 - (k+1)*x)). - Michael Somos, Feb 26 2014
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 3) / LambertW(n)^(n + 7/2). - Vaclav Kotesovec, Jun 10 2020
a(0) = 1; a(n) = 3 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020
a(n) = Sum_{k=0..n} 4^k*A124323(n, k). - Mélika Tebni, Jun 10 2022

A049020 Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 10, 6, 1, 15, 37, 31, 10, 1, 52, 151, 160, 75, 15, 1, 203, 674, 856, 520, 155, 21, 1, 877, 3263, 4802, 3556, 1400, 287, 28, 1, 4140, 17007, 28337, 24626, 11991, 3290, 490, 36, 1, 21147, 94828, 175896, 174805, 101031, 34671, 6972, 786, 45, 1

Offset: 0

N. J. A. Sloane

Triangle a(n,k) read by rows; given by [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Deléham's operator defined in A084938.
Exponential Riordan array [exp(exp(x)-1), exp(x)-1]. - Paul Barry, Jan 12 2009
Equal to A048993*A007318. - Philippe Deléham, Oct 31 2011
This lower unitriangular array is the L factor in the LU decomposition of the Hankel matrix (Bell(i+j-2))A000110(n).%20The%20U%20factor%20is%20A059098%20(see%20Chamberland,%20p.%201672).%20-%20_Peter%20Bala">i,j >= 1, where Bell(n) = A000110(n). The U factor is A059098 (see Chamberland, p. 1672). - _Peter Bala, Oct 15 2023

			Triangle begins:
   1;
   1,  1;
   2,  3,  1;
   5, 10,  6,  1;
  15, 37, 31, 10,  1;
  ...
From _Paul Barry_, Jan 12 2009: (Start)
Production array begins
  1, 1;
  1, 2, 1;
  0, 2, 3, 1;
  0, 0, 3, 4, 1;
  0, 0, 0, 4, 5, 1;
  ... (End)

Alois P. Heinz, Rows n = 0..140, flattened
M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 11.
Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 15 Feb. 2013, pp. 1667-1677.
J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.
Tom Halverson and Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.
Aoife Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16.
J. Riordan, Letter, Oct 31 1977. The array is on the second page.

First column gives A000110, second column = A005493.
Third column = A003128, row sums = A001861, A059340.
See A244489 for another version of this triangle.
Cf. A059098, A245109, A046716.

a:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, a(n-1, k-1) +(k+1)*(a(n-1, k) +a(n-1, k+1))))
    end:
seq(seq(a(n, k), k=0..n), n=0..15);  # Alois P. Heinz, Nov 30 2012

a[n_, k_] = Sum[StirlingS2[n, i]*Binomial[i, k], {i, 0, n}]; Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]]
(* Jean-François Alcover, Aug 29 2011, after Vladeta Jovovic *)

T(n,k)=if(k<0 || k>n,0,n!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),n),k))

def A049020_triangle(dim):
    M = matrix(ZZ, dim, dim)
    for n in (0..dim-1): M[n, n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n, k] = M[n-1, k-1]+(k+1)*M[n-1, k]+(k+1)*M[n-1, k+1]
    return M
A049020_triangle(9) # Peter Luschny, Sep 19 2012

a(n,k) = a(n-1, k-1) + (k+1)*a(n-1, k) + (k+1)*a(n-1, k+1), n >= 1.
a(n,k) = Sum_{i=0..n} Stirling2(n,i)*binomial(i,k). - Vladeta Jovovic, Jan 27 2001
E.g.f. for the k-th column is (1/k!)*(exp(x)-1)^k*exp(exp(x)-1). - Vladeta Jovovic, Jan 27 2001
G.f.: 1/(1-x-xy-x^2(1+y)/(1-2x-xy-2x^2(1+y)/(1-3x-xy-3x^2(1+y)/(1-4x-xy-4x^2(1+y)/(1-... (continued fraction). - Paul Barry, Apr 29 2009
E.g.f.: exp((y+1)*(exp(x)-1)). - Geoffrey Critzer, Nov 30 2012
Note that A244489 (which is essentially the same triangle) has the formula T(n,k) = Sum_{j=k..n} binomial(n,j)*Stirling_2(j,k)*Bell(n-j), where Bell(n) = A000110(n), for n >= 1, 0 <= k <= n-1. - N. J. A. Sloane, May 17 2016
a(2n,n) = A245109(n). - Alois P. Heinz, Aug 23 2017
Empirical: a(n,k) = p(1^n)[st(1^k)] (see A002872 for notation). - Andrey Zabolotskiy, Oct 17 2017
a(n,k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j)/k!. - Peter Luschny, Dec 06 2023

More terms from James Sellers.

Better definition from Geoffrey Critzer, Nov 30 2012.

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

A003128 Number of driving-point impedances of an n-terminal network.

Original entry on oeis.org

0, 0, 1, 6, 31, 160, 856, 4802, 28337, 175896, 1146931, 7841108, 56089804, 418952508, 3261082917, 26403700954, 221981169447, 1934688328192, 17454004213180, 162765041827846, 1566915224106221, 15553364227949564, 159004783733999787, 1672432865100333916

Offset: 0

N. J. A. Sloane

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

N. J. A. Sloane, Table of n, a(n) for n = 0..100
J. Riordan, The number of impedances of an n-terminal network, Bell Syst. Tech. J., 18 (1939), 300-314.
R. Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

Cf. A000110, A000217, A003129, A003130, A005493, A039759, A039765, A123158.

a003128 n = a003128_list !! n
a003128_list = zipWith3 (\x y z -> (x - 3 * y + z) `div` 2)
               a000110_list (tail a000110_list) (drop 2 a000110_list)
-- Reinhard Zumkeller, Jun 30 2013

[(Bell(n) - 3*Bell(n+1) + Bell(n+2))/2: n in [0..30]]; // Vincenzo Librandi, Sep 19 2014

with(combinat); A000110:=n->sum(stirling2(n, k), k=0..n): f:=n->(A000110(n)-3*A000110(n+1)+A000110(n+2))/2;

a[n_] := (BellB[n] - 3*BellB[n+1] + BellB[n+2])/2; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 12 2012, after Vladeta Jovovic *)
max = 23; CoefficientList[ Series[1/2*(E^x - 1)^2*E^(E^x - 1), {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Oct 04 2013, after e.g.f. *)

makelist((belln(n)-3*belln(n+1)+belln(n+2))/2,n,0,23); /* Emanuele Munarini, Jul 14 2011 */

a(n)=sum(k=1,n,binomial(k,2)*stirling(n,k,2)) \\ Charles R Greathouse IV, Feb 07 2017

# Python 3.2 or higher required
from itertools import accumulate
A003128_list, blist, a, b = [], [1], 1, 1
for _ in range(30):
    blist = list(accumulate([b]+blist))
    c = blist[-1]
    A003128_list.append((c+a-3*b)//2)
    a, b = b, c # Chai Wah Wu, Sep 19 2014

def A003128(n): return (bell_number(n) - 3*bell_number(n+1) + bell_number(n+2))/2
[A003128(n) for n in range(40)] # G. C. Greubel, Nov 04 2022

a(n) = (Bell(n) - 3*Bell(n+1) + Bell(n+2))/2. - Vladeta Jovovic, Aug 07 2006
a(n+2) = A123158(n,4). - Philippe Deléham, Oct 06 2006
From Peter Bala, Nov 28 2011: (Start)
a(n) = Sum_{k=1..n} binomial(k,2)*Stirling2(n,k), Stirling transform of A000217.
a(n) = (1/(2*exp(1)))*Sum_{k>=0} k^n*(k^2-3*k+1)/k!. Note that k^2-3*k+1 = k*(k-1)-2*k+1 is an example of a Poisson-Charlier polynomial.
a(n) = D^n(x^2/2!*exp(x)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A005493.
E.g.f.: (1/2)*exp(exp(x)-1)*(exp(x)-1)^2 = x^2/2! + 6*x^3/3! + 31*x^4/4! + ...
O.g.f.: Sum_{k>=0} binomial(k,2)*x^k/Product_{i=1..k} (1-i*x) = x^2 + 6*x^3 + 31*x^4 + ... (End)
a(n) ~ n^2 * Bell(n) / (2*LambertW(n)^2) * (1 - 3*LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021

More terms from Vladeta Jovovic, Apr 14 2000
Typo in entries corrected by Martin Larsen, Jul 03 2008
Typo in e.g.f. corrected by Vaclav Kotesovec, Feb 15 2015

A011965 Second differences of Bell numbers.

Original entry on oeis.org

1, 2, 7, 27, 114, 523, 2589, 13744, 77821, 467767, 2972432, 19895813, 139824045, 1028804338, 7905124379, 63287544055, 526827208698, 4551453462543, 40740750631417, 377254241891064, 3608700264369193, 35613444194346451, 362161573323083920, 3790824599495473121

Offset: 0

N. J. A. Sloane

Number of partitions of n+3 with at least one singleton and with the smallest element in a singleton equal to 3. Alternatively, number of partitions of n+3 with at least one singleton and with the largest element in a singleton equal to n+1. - Olivier GERARD, Oct 29 2007

Out of the A005493(n) set partitions with a specific two elements clustered separately, number that have a different set of two elements clustered separately. - Andrey Goder (andy.goder(AT)gmail.com), Dec 17 2007

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

Chai Wah Wu, Table of n, a(n) for n = 0..570(terms 0..250 from Alois P. Heinz).
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]
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Cf. A000110, A005493, A106436.

Similar recurrences: A006014, A090365, A124758, A217924, A243499, A284005, A290615, A329369, A341392, A347204, A347205, A350309.

[Bell(n+2) -2*Bell(n+1) + Bell(n): n in [0..40]]; // G. C. Greubel, Jan 07 2025

a:= n-> add((-1)^k*binomial(2,k)*combinat['bell'](n+k), k=0..2): seq(a(n), n=0..20);  # Alois P. Heinz, Sep 05 2008

Differences[BellB[Range[0, 30]], 2] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011965_list, blist, b = [1], [1, 2], 2
for _ in range(1000):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    A011965_list.append(blist[-3])
# Chai Wah Wu, Sep 02 2014

# or Sagemath
b=bell_number
print([b(n+2) -2*b(n+1) +b(n) for n in range(41)]) # G. C. Greubel, Jan 07 2025

a(n) = A005493(n) - A005493(n-1).
E.g.f.: exp(exp(x)-1)*(exp(2*x)-exp(x)+1). - Vladeta Jovovic, Feb 11 2003
a(n) = A000110(n) - 2*A000110(n-1) + A000110(n-2). - Andrey Goder (andy.goder(AT)gmail.com), Dec 17 2007
G.f.: G(0) 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+3*x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012
G.f.: 1 - G(0) where G(k) = 1 - 1/(1-k*x-2*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 17 2013
G.f.: 1 - 1/x + (1-x)^2/x/(G(0)-x) where G(k) = 1 - x*(k+1)/(1 - x/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
G.f.: G(0)*(1-1/x) where G(k) = 1 - 1/(1-x*(k+1))/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 07 2013
a(n) ~ n^2 * Bell(n) / LambertW(n)^2 * (1 - 2*LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021
Conjecture: a(n) = Sum_{k=0..2^n - 1} b(k) for n >= 0 where b(2n+1) = b(n) + b(A025480(n-1)), b(2n) = b(n - 2^f(n)) + b(2n - 2^f(n)) + b(A025480(n-1)) for n > 0 with b(0) = b(1) = 1 and where f(n) = A007814(n). Also b((4^n - 1)/3) = A141154(n+1). - Mikhail Kurkov, Jan 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

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

A003436 Number of inequivalent labeled Hamiltonian circuits on n-octahedron. Interlacing chords joining 2n points on circle.

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A278990 Number of loopless linear chord diagrams with n chords.

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A005494 3-Bell numbers: E.g.f.: exp(3*z + exp(z) - 1).

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A049020 Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.

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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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A003128 Number of driving-point impedances of an n-terminal network.

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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).