A005494 -id:A005494 - cp's OEIS frontend

A000178 Superfactorials: product of first n factorials.

1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, 1834933472251084800000, 6658606584104736522240000000, 265790267296391946810949632000000000, 127313963299399416749559771247411200000000000, 792786697595796795607377086400871488552960000000000000

Offset: 0

N. J. A. Sloane

a(n) is also the Vandermonde determinant of the numbers 1,2,...,(n+1), i.e., the determinant of the (n+1) X (n+1) matrix A with A[i,j] = i^j, 1 <= i <= n+1, 0 <= j <= n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001
a(n) = (1/n!) * D(n) where D(n) is the determinant of order n in which the (i,j)-th element is i^j. - Amarnath Murthy, Jan 02 2002
Determinant of S_n where S_n is the n X n matrix S_n(i,j) = Sum_{d|i} d^j. - Benoit Cloitre, May 19 2002
Appears to be det(M_n) where M_n is the n X n matrix with m(i,j) = J_j(i) where J_k(n) denote the Jordan function of row k, column n (cf. A059380(m)). - Benoit Cloitre, May 19 2002
a(2n+1) = 1, 12, 34560, 125411328000, ... is the Hankel transform of A000182 (tangent numbers) = 1, 2, 16, 272, 7936, ...; example: det([1, 2, 16, 272; 2, 16, 272, 7936; 16, 272, 7936, 353792; 272, 7936, 353792, 22368256]) = 125411328000. - Philippe Deléham, Mar 07 2004
Determinant of the (n+1) X (n+1) matrix whose i-th row consists of terms 1 to n+1 of the Lucas sequence U(i,Q), for any Q. When Q=0, the Vandermonde matrix is obtained. - T. D. Noe, Aug 21 2004
Determinant of the (n+1) X (n+1) matrix A whose elements are A(i,j) = B(i+j) for 0 <= i,j <= n, where B(k) is the k-th Bell number, A000110(k) [I. Mezo, JIS 14 (2011) # 11.1.1]. - T. D. Noe, Dec 04 2004
The Hankel transform of the sequence A090365 is A000178(n+1); example: det([1,1,3; 1,3,11; 3,11,47]) = 12. - Philippe Deléham, Mar 02 2005
Theorem 1.3, page 2, of Polynomial points, Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6, provides an example of an Abelian quotient group of order (n-1) superfactorial, for each positive integer n. The quotient is obtained from sequences of polynomial values. - E. F. Cornelius, Jr. (efcornelius(AT)comcast.net), Apr 09 2007
Starting with offset 1 this is a 'Matryoshka doll' sequence with alpha=1, the multiplicative counterpart to the additive A000292. seq(mul(mul(i,i=alpha..k), k=alpha..n),n=alpha..12). - Peter Luschny, Jul 14 2009
For n>0, a(n) is also the determinant of S_n where S_n is the n X n matrix, indexed from 1, S_n(i,j)=sigma_i(j), where sigma_k(n) is the generalized divisor sigma function: A000203 is sigma_1(n). - Enrique Pérez Herrero, Jun 21 2010
a(n) is the multiplicative Wiener index of the (n+1)-vertex path. Example: a(4)=288 because in the path on 5 vertices there are 3 distances equal to 2, 2 distances equal to 3, and 1 distance equal to 4 (2*2*2*3*3*4=288). See p. 115 of the Gutman et al. reference. - Emeric Deutsch, Sep 21 2011
a(n-1) = Product_{j=1..n-1} j! = V(n) = Product_{1 <= i < j <= n} (j - i) (a Vandermondian V(n), see the Ahmed Fares May 06 2001 comment above), n >= 1, is in fact the determinant of any n X n matrix M(n) with entries M(n;i,j) = p(j-1,x = i), 1 <= i, j <= n, where p(m,x), m >= 0, are monic polynomials of exact degree m with p(0,x) = 1. This is a special x[i] = i choice in a general theorem given in Vein-Dale, p. 59 (written for the transposed matrix M(n;j,x_i) = p(i-1,x_j) = P_i(x_j) in Vein-Dale, and there a_{k,k} = 1, for k=1..n). See the Aug 26 2013 comment under A049310, where p(n,x) = S(n,x) (Chebyshev S). - Wolfdieter Lang, Aug 27 2013
a(n) is the number of monotonic magmas on n elements labeled 1..n with a symmetric multiplication table. I.e., Product(i,j) >= max(i,j); Product(i,j) = Product(j,i). - Chad Brewbaker, Nov 03 2013
The product of the pairwise differences of n+1 integers is a multiple of a(n) [and this does not hold for any k > a(n)]. - Charles R Greathouse IV, Aug 15 2014
a(n) is the determinant of the (n+1) X (n+1) matrix M with M(i,j) = (n+j-1)!/(n+j-i)!, 1 <= i <= n+1, 1 <= j <= n+1. - Stoyan Apostolov, Aug 26 2014
All terms are in A064807 and all terms after a(2) are in A005101. - Ivan N. Ianakiev, Sep 02 2016
Empirical: a(n-1) is the determinant of order n in which the (i,j)-th entry is the (j-1)-th derivative of x^(x+i-1) evaluated at x=1. - John M. Campbell, Dec 13 2016
Empirical: If f(x) is a smooth, real-valued function on an open neighborhood of 0 such that f(0)=1, then a(n) is the determinant of order n+1 in which the (i,j)-th entry is the (j-1)-th derivative of f(x)/((1-x)^(i-1)) evaluated at x=0. - John M. Campbell, Dec 27 2016
Also the automorphism group order of the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
Is the zigzag Hankel transform of A000182. That is, a(2*n+1) is the Hankel transform of A000182 and a(2*n+2) is the Hankel transform of A000182(n+1). - Michael Somos, Mar 11 2020
Except for n = 0, 1, superfactorial a(n) is never a square (proof in link Mabry and Cormick, FFF 4 p. 349); however, when k belongs to A349079 (see for further information), there exists m, 1 <= m <= k such that a(k) / m! is a square. - Bernard Schott, Nov 29 2021

			a(3) = (1/6)* | 1 1 1 | 2 4 8 | 3 9 27 |
a(7) = n! * a(n-1) = 7! * 24883200 = 125411328000.
a(12) = 1! * 2! * 3! * 4! * 5! * 6! * 7! * 8! * 9! * 10! * 11! * 12!
= 1^12 * 2^11 * 3^10 * 4^9 * 5^8 * 6^7 * 7^6 * 8^5 * 9^4 * 10^3 * 11^2 * 12^1
= 2^56 * 3^26 * 5^11 * 7^6 * 11^2.
G.f. = 1 + x + 2*x^2 + 12*x^3 + 288*x^4 + 34560*x^5 + 24883200*x^6 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 545.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 231.
H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 53.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer, 1999.

Boris Hostnik, Table of n, a(n) for n = 0..46
Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443.
Andreas B. G. Blobel, On convolution powers of 1/x, arXiv:2203.09519 [math.CO], 2022.
E. F. Cornelius, Jr. and Phill Schultz, Polynomial points , Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6.
Selden Crary, Factorization of the Determinant of the Gaussian-Covariance Matrix of Evenly Spaced Points Using an Inter-dimensional Multiset Duality, arXiv preprint arXiv:1406.6326 [math.ST], 2014-2019.
N. Destainville, R. Mosseri and F. Bailly, Configurational Entropy of Codimension-One Tilings and Directed Membranes, J. Stat. Phys. 87, Nos 3/4, 697 (1997).
J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.
Richard Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, 107 (2000), 557-560.
William Q. Erickson and Jan Kretschmann, The structure and normalized volume of Monge polytopes, arXiv:2311.07522 [math.CO], 2023. See p. 7.
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [Broken link]
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [From the Wayback machine]
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Ivan Gutman, Wolfgang Linert, István Lukovits and Željko Tomović, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., Vol. 40, No. 1 (2000), pp. 113-116.
Brady Haran and Sophie Maclean, What's special about 288?, Numberphile video (2023).
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Nick Hobson, Python program for this sequence.
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
Pavel L. Krapivsky, Jean-Marc Luck and Kirone Mallick, Quantum return probability of a system of N non-interacting lattice fermions, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, No. 2 (2018), 023104; arXiv preprint, arXiv:1710.08178 [cond-mat.mes-hall], 2017-2018.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, International Journal of Number Theory, Vol. 12, No. 1 (2016), pp. 57-91; arXiv preprint, arXiv:1409.4145 [math.NT], 2014-2015.
Mogens Esrom Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.
John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 2009.
Rémy Mosseri and Francis Bailly, Configurational Entropy in Octagonal Tiling Models, Int. J. Mod. Phys. B, Vol. 7, No. 6-7 (1993), pp. 1427-1436.
Rémy Mosseri, F. Bailly and C. Sire, Configurational Entropy in Random Tiling Models, J. Non-Cryst. Solids, Vol. 153-154 (1993), pp. 201-204.
Amarnath Murthy, Miscellaneous Results and Theorems on Smarandache terms and factor partitions, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 3.14.
Christian Radoux, Query 145, Notices Amer. Math. Soc., 25-3 (1978), p. 197.
Christian Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
Vignesh Raman, The Generalized Superfactorial, Hyperfactorial and Primorial functions, arXiv:2012.00882 [math.NT], 2020.
Michel Waldschmidt, Schanuel Property for Elliptic and Quasi-Elliptic Functions, arXiv:2504.14041 [math.NT], 2025. Mentions this sequence, see p. 10.
Eric Weisstein's World of Mathematics, Barnes G-Function.
Eric Weisstein's World of Mathematics, Bell Number.
Eric Weisstein's World of Mathematics, Factorial Products.
Eric Weisstein's World of Mathematics, Graph Automorphism.
Eric Weisstein's World of Mathematics, Lucas Sequence.
Eric Weisstein's World of Mathematics, Superfactorial.
Eric Weisstein's World of Mathematics, Vandermonde Determinant.
Index to divisibility sequences.
Index entries for sequences related to factorial numbers.
Index to sequences related to Olympiads and other Mathematical competitions.

Partial products of A000142.
Cf. A002109, A036561, A000292, A098694, A098695, A113271, A087316, A113208, A113231, A113257, A113258, A113320, A113336, A113498, A113173, A113170, A113475, A113492, A113497, A113533, A113534, A113535, A113153, A113154, A113122, A114045, A055462, A137986, A137987.
A002109(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
A000178 is the Hankel transform (see A001906 for definition) of A000085, A000110, A000296, A005425, A005493, A005494 and A045379. - John W. Layman, Jul 28 2000
Cf. A255322, A255358, A255359, A255360.
Cf. A051675, A255321, A255323, A255344, A287013.
Cf. A348692, A349079.

[&*[Factorial(k): k in [0..n]]: n in [0..20]]; // Bruno Berselli, Mar 11 2015

A000178 := proc(n)
    mul(i!,i=1..n) ;
end proc:
seq(A000178(n),n=0..10) ; # R. J. Mathar, Oct 30 2015

a[0] := 1; a[1] := 1; a[n_] := n!*a[n - 1]; Table[a[n], {n, 1, 12}] (* Stefan Steinerberger, Mar 10 2006 *)
Table[BarnesG[n], {n, 2, 14}] (* Zerinvary Lajos, Jul 16 2009 *)
FoldList[Times,1,Range[20]!] (* Harvey P. Dale, Mar 25 2011 *)
RecurrenceTable[{a[n] == n! a[n - 1], a[0] == 1}, a, {n, 0, 12}] (* Ray Chandler, Jul 30 2015 *)
BarnesG[Range[2, 20]] (* Eric W. Weisstein, Jul 14 2017 *)

A000178(n):=prod(k!,k,0,n)$ makelist(A000178(n),n,0,30); /* Martin Ettl, Oct 23 2012 */

A000178(n)=prod(k=2,n,k!) \\ M. F. Hasler, Sep 02 2007

a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, (1+j!*x+x*O(x^n)) )), n) \\ Paul D. Hanna, Oct 02 2013

for(j=1,13, print1(prod(k=1,j,k^(j-k)),", ")) \\ Hugo Pfoertner, Apr 09 2020

A000178_list, n, m = [1], 1,1
for i in range(1,100):
    m *= i
    n *= m
    A000178_list.append(n) # Chai Wah Wu, Aug 21 2015

from math import prod
def A000178(n): return prod(i**(n-i+1) for i in range(2,n+1)) # Chai Wah Wu, Nov 26 2023

def mono_choices(a,b,n)
    n - [a,b].max
end
def comm_mono_choices(n)
    accum =1
    0.upto(n-1) do |i|
        i.upto(n-1) do |j|
            accum = accum * mono_choices(i,j,n)
        end
    end
    accum
end
1.upto(12) do |k|
    puts comm_mono_choices(k)
end # Chad Brewbaker, Nov 03 2013

a(0) = 1, a(n) = n!*a(n-1). - Lee Hae-hwang, May 13 2003, corrected by Ilya Gutkovskiy, Jul 30 2016
a(0) = 1, a(n) = 1^n * 2^(n-1) * 3^(n-2) * ... * n = Product_{r=1..n} r^(n-r+1). - Amarnath Murthy, Dec 12 2003 [Formula corrected by Derek Orr, Jul 27 2014]
a(n) = sqrt(A055209(n)). - Philippe Deléham, Mar 07 2004
a(n) = Product_{i=1..n} Product_{j=0..i-1} (i-j). - Paul Barry, Aug 02 2008
log a(n) = 0.5*n^2*log n - 0.75*n^2 + O(n*log n). - Charles R Greathouse IV, Jan 13 2012
Asymptotic: a(n) ~ exp(zeta'(-1) - 3/4 - (3/4)*n^2 - (3/2)*n)*(2*Pi)^(1/2 + (1/2)*n)*(n+1)^((1/2)*n^2 + n + 5/12). For example, a(100) is approx. 0.270317...*10^6941. (See A213080.) - Peter Luschny, Jun 23 2012
G.f.: 1 + x/(U(0) - x) where U(k) = 1 + x*(k+1)! - x*(k+2)!/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 02 2012
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 1/(1 + 1/((k+1)!*x*G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k!*x). - Paul D. Hanna, Oct 02 2013
A203227(n+1)/a(n) -> e, as n -> oo. - Daniel Suteu, Jul 30 2016
From Ilya Gutkovskiy, Jul 30 2016: (Start)
a(n) = G(n+2), where G(n) is the Barnes G-function.
a(n) ~ exp(1/12 - n*(3*n+4)/4)*n^(n*(n+2)/2 + 5/12)*(2*Pi)^((n+1)/2)/A, where A is the Glaisher-Kinkelin constant (A074962).
Sum_{n>=0} (-1)^n/a(n) = A137986. (End)
0 = a(n)*a(n+2)^3 + a(n+1)^2*a(n+2)^2 - a(n+1)^3*a(n+3) for all n in Z (if a(-1)=1). - Michael Somos, Mar 11 2020
Sum_{n>=0} 1/a(n) = A287013 = 1/A137987. - Amiram Eldar, Nov 19 2020
a(n) = Wronskian(1, x, x^2, ..., x^n). - Mohammed Yaseen, Aug 01 2023
From Andrea Pinos, Apr 04 2024: (Start)
a(n) = e^(Sum_{k=1..n} (Integral_{x=1..k+1} Psi(x) dx)).
a(n) = e^(Integral_{x=1..n+1} (log(sqrt(2*Pi)) - (x-1/2) + x*Psi(x)) dx).
a(n) = e^(Integral_{x=1..n+1} (log(sqrt(2*Pi)) - (x-1/2) + (n+1)*Psi(x) - log(Gamma(x))) dx).
Psi(x) is the digamma function. (End)

A005493 2-Bell numbers: a(n) = number of partitions of [n+1] with a distinguished block.

Original entry on oeis.org

1, 3, 10, 37, 151, 674, 3263, 17007, 94828, 562595, 3535027, 23430840, 163254885, 1192059223, 9097183602, 72384727657, 599211936355, 5150665398898, 45891416030315, 423145657921379, 4031845922290572, 39645290116637023, 401806863439720943, 4192631462935194064

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of Boolean sublattices of the Boolean lattice of subsets of {1..n}.
a(n) = p(n+1) where p(x) is the unique degree n polynomial such that p(k) = A000110(k+1) for k = 0, 1, ..., n. - Michael Somos, Oct 07 2003
With offset 1, number of permutations beginning with 12 and avoiding 21-3.
Rows sums of Bell's triangle (A011971). - Jorge Coveiro, Dec 26 2004
Number of blocks in all set partitions of an (n+1)-set. Example: a(2)=10 because the set partitions of {1,2,3} are 1|2|3, 1|23, 12|3, 13|2 and 123, with a total of 10 blocks. - Emeric Deutsch, Nov 13 2006
Number of partitions of n+3 with at least one singleton and with the smallest element in a singleton equal to 2. - Olivier Gérard, Oct 29 2007
See page 29, Theorem 5.6 of my paper on the arXiv: These numbers are the dimensions of the homogeneous components of the operad called ComTrip associated with commutative triplicial algebras. (Triplicial algebras are related to even trees and also to L-algebras, see A006013.) - Philippe Leroux, Nov 17 2007
Number of set partitions of (n+2) elements where two specific elements are clustered separately. Example: a(1)=3 because 1/2/3, 1/23, 13/2 are the 3 set partitions with 1, 2 clustered separately. - Andrey Goder (andy.goder(AT)gmail.com), Dec 17 2007
Equals A008277 * [1,2,3,...], i.e., the product of the Stirling number of the second kind triangle and the natural number vector. a(n+1) = row sums of triangle A137650. - Gary W. Adamson, Jan 31 2008
Prefaced with a "1" = row sums of triangle A152433. - Gary W. Adamson, Dec 04 2008
Equals row sums of triangle A159573. - Gary W. Adamson, Apr 16 2009
Number of embedded coalitions in an (n+1)-person game. - David Yeung (wkyeung(AT)hkbu.edu.hk), May 08 2008
If prefixed with 0, gives first differences of Bell numbers A000110 (cf. A106436). - N. J. A. Sloane, Aug 29 2013
Sum_{n>=0} a(n)/n! = e^(e+1) = 41.19355567... (see A235214). Contrast with e^(e-1) = Sum_{n>=0} A000110(n)/n!. - Richard R. Forberg, Jan 05 2014

			For example, a(1) counts (12), (1)-2, 1-(2) where dashes separate blocks and the distinguished block is parenthesized.

Olivier Gérard and Karol A. Penson, A budget of set partition statistics, in preparation. Unpublished as of 2017.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Chai Wah Wu, Table of n, a(n) for n = 0..574 (first 101 terms from T. D. Noe)
V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
M. Brookes, J. East, C. Miller, J.D. Mitchell, and N. Ruskuc, Heights of one- and two-sided congruence lattices of semigroups, arXiv:2310.08229 [math.GR], 2023.
Giulio Cerbai, Modified ascent sequences and Bell numbers, arXiv:2305.10820 [math.CO], 2023. See p. 11.
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]
R. B. Corcino and C. B. Corcino, On generalized Bell polynomials, Discrete Dyn. Nat. Soc. Article ID: 623456 (2011).
C. B. Corcino and R. B. Corcino, An asymptotic formula for r-Bell numbers with real arguments, ISRN Discrete Math, 2013 (2013), Article ID 274697.
Gábor Czédli, Lattices with lots of congruence energy, arXiv:2205.02294 [math.RA], 2022.
A. Dil and V. Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series II, Appl. Anal. Discrete Math. 5 (2011), 212-229.
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
A. L. L. Gao, S. Kitaev, and P. B. Zhang, On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
S. Getu et al., How to guess a generating function, SIAM J. Discrete Math., 5 (1992), 497-499.
S. K. Ghosal and J. K. Mandal, Stirling Transform Based Color Image Authentication, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104.
Alain Hertz, Hadrien Mélot, Sébastien Bonte, Gauvain Devillez, and Pierre Hauweele, Upper bounds on the average number of colors in the non-equivalent colorings of a graph, arXiv:2105.01120 [math.CO], 2021.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 152
R. Jakimczuk, Successive Derivatives and Integer Sequences, J. Int. Seq. 14 (2011) # 11.7.3.
S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Philippe Leroux, L-algebras, triplicial-algebras, within an equivalence of categories motivated by graphs (formerly "An equivalence of categories motivated by weighted directed graphs"), arXiv:0709.3453 [math-ph], 2007-2008.
Toufik Mansour and Mark Shattuck, A recurrence related to the Bell numbers, INTEGERS 11 (2011), #A67.
I. Mezo, The r-Bell numbers, J. Integer Seq. 14(1) (2011), Article 11.1.1.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Example 12.16 (pdf, ps)
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
J. Riordan, Letter, Oct 31 1977
Eric Weisstein's World of Mathematics, Stirling Transform.
Eric Weisstein's World of Mathematics, Bell Triangle
E. G. Whitehead, Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.
David W. K. Yeung, Recursive sequence identifying the number of embedded coalitions, International Game Theory Review 10 (1) (2008), 129-136.

Cf. A000110, A005494, A008277, A011971, A011968, A049020, A106436, A124323, A137650, A152433, A159573.
A row or column of the array A108087.
Row sums of triangle A143494. - Wolfdieter Lang, Sep 29 2011. And also of triangle A362924. - N. J. A. Sloane, Aug 10 2023

with(combinat): seq(bell(n+2)-bell(n+1),n=0..22); # Emeric Deutsch, Nov 13 2006
seq(add(binomial(n, k)*(bell(n-k)), k=1..n), n=1..23); # Zerinvary Lajos, Dec 01 2006
A005493  := proc(n) local a,b,i;
a := [seq(3,i=1..n)]; b := [seq(2,i=1..n)];
2^n*exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=1,%),66)) end:
seq(A005493(n),n=0..22); # Peter Luschny, Mar 30 2011
BT := proc(n,k) option remember; if n = 0 and k = 0 then 1
elif k = n then BT(n-1,0) else BT(n,k+1)+BT(n-1,k) fi end:
A005493 := n -> add(BT(n,k),k=0..n):
seq(A005493(i),i=0..22); # Peter Luschny, Aug 04 2011
# For Maple code for r-Bell numbers, etc., see A232472. - N. J. A. Sloane, Nov 27 2013

a=Exp[x]-1; Rest[CoefficientList[Series[a Exp[a],{x,0,20}],x] * Table[n!,{n,0,20}]]
a[ n_] := If[ n<0, 0, With[ {m = n+1}, m! SeriesCoefficient[ # Exp@# &[ Exp@x - 1], {x, 0, m}]]]; (* Michael Somos, Nov 16 2011 *)
Differences[BellB[Range[30]]] (* Harvey P. Dale, Oct 16 2014 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) + 2*x - 1), n))}; /* Michael Somos, Oct 09 2002 */

{a(n) = if( n<0, 0, n+=2; subst( polinterpolate( Vec( serlaplace( exp( exp( x + O(x^n)) - 1) - 1))), x, n))}; /* Michael Somos, Oct 07 2003 */

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A005493_list, blist, b = [], [1], 1
for _ in range(1001):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    A005493_list.append(blist[-2])
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

a(n-1) = Sum_{k=1..n} k*Stirling2(n, k) for n>=1.
E.g.f.: exp(exp(x) + 2*x - 1). First differences of Bell numbers (if offset 1). - Michael Somos, Oct 09 2002
G.f.: Sum_{k>=0} (x^k/Product_{l=1..k} (1-(l+1)x)). - Ralf Stephan, Apr 18 2004
a(n) = Sum_{i=0..n} 2^(n-i)*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n). - Fred Lunnon, Aug 04 2007 [Written umbrally, a(n) = (B+2)^n. - N. J. A. Sloane, Feb 07 2009]
Representation as an infinite series: a(n-1) = Sum_{k>=2} (k^n*(k-1)/k!)/exp(1), n=1, 2, ... This is a Dobinski-type summation formula. - Karol A. Penson, Mar 14 2002
Row sums of A011971 (Aitken's array, also called Bell triangle). - Philippe Deléham, Nov 15 2003
a(n) = exp(-1)*Sum_{k>=0} ((k+2)^n)/k!. - Gerald McGarvey, Jun 03 2004
Recurrence: a(n+1) = 1 + Sum_{j=1..n} (1+binomial(n, j))*a(j). - Jon Perry, Apr 25 2005
a(n) = A000296(n+3) - A000296(n+1). - Philippe Deléham, Jul 31 2005
a(n) = B(n+2) - B(n+1), where B(n) are Bell numbers (A000110). - Franklin T. Adams-Watters, Jul 13 2006
a(n) = A123158(n,2). - Philippe Deléham, Oct 06 2006
Binomial transform of Bell numbers 1, 2, 5, 15, 52, 203, 877, 4140, ... (see A000110).
Define f_1(x), f_2(x), ... such that f_1(x)=x*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
Representation of numbers a(n), n=0,1..., as special values of hypergeometric function of type (n)F(n), in Maple notation: a(n)=exp(-1)*2^n*hypergeom([3,3...3],[2.2...2],1), n=0,1..., i.e., having n parameters all equal to 3 in the numerator, having n parameters all equal to 2 in the denominator and the value of the argument equal to 1. Examples: a(0)= 2^0*evalf(hypergeom([],[],1)/exp(1))=1 a(1)= 2^1*evalf(hypergeom([3],[2],1)/exp(1))=3 a(2)= 2^2*evalf(hypergeom([3,3],[2,2],1)/exp(1))=10 a(3)= 2^3*evalf(hypergeom([3,3,3],[2,2,2],1)/exp(1))=37 a(4)= 2^4*evalf(hypergeom([3,3,3,3],[2,2,2,2],1)/exp(1))=151 a(5)= 2^5*evalf(hypergeom([3,3,3,3,3],[2,2,2,2,2],1)/exp(1)) = 674. - Karol A. Penson, Sep 28 2007
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
a(n) = D^(n+1)(x*exp(x)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A003128, A052852 and A009737. - Peter Bala, Nov 25 2011
From Sergei N. Gladkovskii, Oct 11 2012 to Jan 26 2014: (Start)
Continued fractions:
G.f.: 1/U(0) where U(k) = 1 - x*(k+3) - x^2*(k+1)/U(k+1).
G.f.: 1/(U(0)-x) where U(k) = 1 - x - x*(k+1)/(1 - x/U(k+1)).
G.f.: G(0)/(1-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+3*x-1)/G(k+1) )).
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-2*x-k*x)/(1-x/(x-1/G(k+1) )).
G.f.: -G(0)/x where G(k) = 1 - 1/(1-k*x-x)/(1-x/(x-1/G(k+1) )).
G.f.: 1 - 2/x + (1/x-1)*S where S = sum(k>=0, ( 1 + (1-x)/(1-x-x*k) )*(x/(1-x))^k / prod(i=0..k-1, (1-x-x*i)/(1-x) ) ).
G.f.: (1-x)/x/(G(0)-x) - 1/x where G(k) = 1 - x*(k+1)/(1 - x/G(k+1) ).
G.f.: (1/G(0) - 1)/x^3 where G(k) = 1 - x/(x - 1/(1 + 1/(x*k-1)/G(k+1) )).
G.f.: 1/Q(0), where Q(k)= 1 - 2*x - x/(1 - x*(k+1)/Q(k+1)).
G.f.: G(0)/(1-3*x), where G(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1 - x*(k+3))*(1 - x*(k+4))/G(k+1) ). (End)
a(n) ~ exp(n/LambertW(n) + 3*LambertW(n)/2 - n - 1) * n^(n + 1/2) / LambertW(n)^(n+1). - Vaclav Kotesovec, Jun 09 2020
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020
a(n) ~ n^2 * Bell(n) / LambertW(n)^2 * (1 - LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021
a(n) = Sum_{k=0..n} 3^k*A124323(n, k). - Mélika Tebni, Jun 02 2022

Definition revised by David Callan, Oct 11 2005

A056857 Triangle read by rows: T(n,c) = number of successive equalities in set partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 6, 3, 1, 15, 20, 12, 4, 1, 52, 75, 50, 20, 5, 1, 203, 312, 225, 100, 30, 6, 1, 877, 1421, 1092, 525, 175, 42, 7, 1, 4140, 7016, 5684, 2912, 1050, 280, 56, 8, 1, 21147, 37260, 31572, 17052, 6552, 1890, 420, 72, 9, 1, 115975, 211470, 186300, 105240, 42630, 13104, 3150, 600, 90, 10, 1

Offset: 1

Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000

Number of successive equalities s_i = s_{i+1} in a set partition {s_1, ..., s_n} of {1, ..., n}, where s_i is the subset containing i, s(1) = 1 and s(i) <= 1 + max of previous s(j)'s.
T(n,c) = number of set partitions of the set {1,2,...,n} in which the size of the block containing the element 1 is k+1. Example: T(4,2)=3 because we have 123|4, 124|3 and 134|2. - Emeric Deutsch, Nov 10 2006
Let P be the lower-triangular Pascal-matrix (A007318), Then this is exp(P) / exp(1). - Gottfried Helms, Mar 30 2007. [This comment was erroneously attached to A011971, but really belongs here. - N. J. A. Sloane, May 02 2015]
From David Pasino (davepasino(AT)yahoo.com), Apr 15 2009: (Start)
As an infinite lower-triangular matrix (with offset 0 rather than 1, so the entries would be B(n - c)*binomial(n, c), B() a Bell number, rather than B(n - 1 - c)*binomial(n - 1, c) as below), this array is S P S^-1 where P is the Pascal matrix A007318, S is the Stirling2 matrix A048993, and S^-1 is the Stirling1 matrix A048994.
Also, S P S^-1 = (1/e)*exp(P). (End)
Exponential Riordan array [exp(exp(x)-1), x]. Equal to A007318*A124323. - Paul Barry, Apr 23 2009
Equal to A049020*A048994 as infinite lower triangular matrices. - Philippe Deléham, Nov 19 2011
Build a superset Q[n] of set partitions of {1,2,...,n} by distinguishing "some" (possibly none or all) of the singletons. Indexed from n >= 0, 0 <= k <= n, T(n,k) is the number of elements in Q[n] that have exactly k distinguished singletons. A singleton is a subset containing one element. T(3,1) = 6 because we have {{1}'{2,3}}, {{1,2}{3}'}, {{1,3}{2}'}, {{1}'{2}{3}}, {{1}{2}'{3}}, {{1}{2}{3}'}. - Geoffrey Critzer, Nov 10 2012
Let Bell(n,x) denote the n-th Bell polynomial, the n-th row polynomial of A048993. Then this is the triangle of connection constants when expressing the basis polynomials Bell(n,x + 1) in terms of the basis polynomials Bell(n,x). For example, row 3 is (5, 6, 3, 1) and 5 + 6*Bell(1,x) + 3*Bell(2,x) + Bell(3,x) = 5 + 6*x + 3*(x + x^2) + (x + 3*x^2 + x^3) = 5 + 10*x + 6*x^2 + x^3 = (x + 1) + 3*(x + 1)^2 + (x + 1)^3 = Bell(3,x + 1). - Peter Bala, Sep 17 2013

			For example {1, 2, 1, 2, 2, 3} is a set partition of {1, 2, 3, 4, 5, 6} and has 1 successive equality, at i = 4.
Triangle begins:
    1;
    1,   1;
    2,   2,   1;
    5,   6,   3,   1;
   15,  20,  12,   4,   1;
   52,  75,  50,  20,   5,   1;
  203, 312, 225, 100,  30,   6,   1;
  ...
From _Paul Barry_, Apr 23 2009: (Start)
Production matrix is
  1,  1;
  1,  1,  1;
  1,  2,  1,  1;
  1,  3,  3,  1,  1;
  1,  4,  6,  4,  1,  1;
  1,  5, 10, 10,  5,  1,  1;
  1,  6, 15, 20, 15,  6,  1,  1;
  1,  7, 21, 35, 35, 21,  7,  1,  1;
  1,  8, 28, 56, 70, 56, 28,  8,  1,  1; ... (End)

W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000. [Apparently unpublished]

Alois P. Heinz, Rows n = 1..141, flattened
H. W. Becker, Rooks and rhymes, Math. Mag., 22 (1948/49), 23-26. See Table III.
H. W. Becker, Rooks and rhymes, Math. Mag., 22 (1948/49), 23-26. [Annotated scanned copy]
Fufa Beyene, Jörgen Backelin, Roberto Mantaci, and Samuel A. Fufa, Set Partitions and Other Bell Number Enumerated Objects, J. Int. Seq., Vol. 26 (2023), Article 23.1.8.
A. Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2, Corollary 17.
G. Hurst and A. Schultz, An elementary (number theory) proof of Touchard's congruence, arXiv:0906.0696v2 [math.CO], 2009.
A. O. Munagi, Set partitions with successions and separations, Intl. J. Math. Math. Sci. 2005 (2005) 451-463.
M. Spivey, A generalized recurrence for Bell numbers, J. Int. Seq., 11 (2008), no. 2, Article 08.2.5
W. Yang, Bell numbers and k-trees, Disc. Math. 156 (1996) 247-252.

Cf. Bell numbers A000110 (column c=0), A052889 (c=1), A105479 (c=2), A105480 (c=3).
Cf. A056858-A056863. Essentially same as A056860, where the rows are read from right to left.
Cf. also A007318, A005493, A270953.
See A259691 for another version.
T(2n+1,n+1) gives A124102.
T(2n,n) gives A297926.

with(combinat): A056857:=(n,c)->binomial(n-1,c)*bell(n-1-c): for n from 1 to 11 do seq(A056857(n,c),c=0..n-1) od; # yields sequence in triangular form; Emeric Deutsch, Nov 10 2006
with(linalg): # Yields sequence in matrix form:
A056857_matrix := n -> subs(exp(1)=1, exponential(exponential(
matrix(n,n,[seq(seq(`if`(j=k+1,j,0),k=0..n-1),j=0..n-1)])))):
A056857_matrix(8); # Peter Luschny, Apr 18 2011

t[n_, k_] := BellB[n-1-k]*Binomial[n-1, k]; Flatten[ Table[t[n, k], {n, 1, 11}, {k, 0, n-1}]](* Jean-François Alcover, Apr 25 2012, after Emeric Deutsch *)

B(n) = sum(k=0, n, stirling(n, k, 2));
tabl(nn)={for(n=1, nn, for(k=0, n - 1, print1(B(n - 1 - k) * binomial(n - 1, k),", ");); print(););};
tabl(12); \\ Indranil Ghosh, Mar 19 2017

from sympy import bell, binomial
for n in range(1,12):
    print([bell(n - 1 - k) * binomial(n - 1, k) for k in range(n)]) # Indranil Ghosh, Mar 19 2017

def a(n): return (-1)^n / factorial(n)
@cached_function
def p(n, m):
    R = PolynomialRing(QQ, "x")
    if n == 0: return R(a(m))
    return R((m + x)*p(n - 1, m) - (m + 1)*p(n - 1, m + 1))
for n in range(11): print(p(n, 0).list())  # Peter Luschny, Jun 18 2023

T(n,c) = B(n-1-c)*binomial(n-1, c), where T(n,c) is the number of set partitions of {1, ..., n} that have c successive equalities and B() is a Bell number.
E.g.f.: exp(exp(x)+x*y-1). - Vladeta Jovovic, Feb 13 2003
G.f.: 1/(1-xy-x-x^2/(1-xy-2x-2x^2/(1-xy-3x-3x^2/(1-xy-4x-4x^2/(1-... (continued fraction). - Paul Barry, Apr 23 2009
Considered as triangle T(n,k), 0 <= k <= n: T(n,k) = A007318(n,k)*A000110(n-k) and Sum_{k=0..n} T(n,k)*x^k = A000296(n), A000110(n), A000110(n+1), A005493(n), A005494(n), A045379(n) for x = -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Dec 13 2009
Let R(n,x) denote the n-th row polynomial of the triangle. Then A000110(n+j) = Bell(n+j,1) = Sum_{k = 1..n} R(j,k)*Stirling2(n,k) (Spivey). - Peter Bala, Sep 17 2013

More terms from David Wasserman, Apr 22 2002

A143495 Triangle read by rows: 3-Stirling numbers of the second kind.

Original entry on oeis.org

1, 3, 1, 9, 7, 1, 27, 37, 12, 1, 81, 175, 97, 18, 1, 243, 781, 660, 205, 25, 1, 729, 3367, 4081, 1890, 380, 33, 1, 2187, 14197, 23772, 15421, 4550, 644, 42, 1, 6561, 58975, 133057, 116298, 47271, 9702, 1022, 52, 1, 19683, 242461, 724260, 830845, 447195

Offset: 3

Peter Bala, Aug 20 2008

This is the case r = 3 of the r-Stirling numbers of the second kind. The 3-Stirling numbers of the second kind give the number of ways of partitioning the set {1,2,...,n} into k nonempty disjoint subsets with the restriction that the elements 1, 2 and 3 belong to distinct subsets. For remarks on the general case see A143494 (r = 2). The corresponding array of 3-Stirling numbers of the first kind is A143492. The theory of r-Stirling numbers of both kinds is developed in [Broder]. For 3-Lah numbers refer to A143498.
From Peter Bala, Sep 19 2008: (Start)
Let D be the derivative operator d/dx and E the Euler operator x*d/dx. Then x^(-3)*E^n*x^3 = Sum_{k = 0..n} T(n+3,k+3)*x^k*D^k.
The row generating polynomials R_n(x) := Sum_{k= 3..n} T(n,k)*x^k satisfy the recurrence R_(n+1)(x) = x*R_n(x) + x*d/dx(R_n(x)) with R_3(x) = x^3. It follows that the polynomials R_n(x) have only real zeros (apply Corollary 1.2. of [Liu and Wang]).
Relation with the 3-Eulerian numbers E_3(n,j) := A144697(n,j): T(n,k) = (3!/k!)*Sum_{j = n-k..n-3} E_3(n,j)*binomial(j,n-k) for n >= k >= 3.
(End)
T(n,k) = S(n,k,3), n>=k>=3, in Mikhailov's first paper, eq.(28) or (A3). E.g.f. column k from (A20) with k->3, r->k. Therefore, with offset [0,0], this triangle is the Sheffer triangle (exp(3*x),exp(x)-1) with e.g.f. of column no. m>=0: exp(3*x)*((exp(x)-1)^m)/m!. See one of the formulas given below. For Sheffer matrices see the W. Lang link under A006232 with the S. Roman reference, also found in A132393. - Wolfdieter Lang, Sep 29 2011

			Triangle begins
  n\k|....3....4....5....6....7....8
  ==================================
  3..|....1
  4..|....3....1
  5..|....9....7....1
  6..|...27...37...12....1
  7..|...81..175...97...18....1
  8..|..243..781..660..205...25....1
  ...
T(5,4) = 7. The set {1,2,3,4,5} can be partitioned into four subsets such that 1, 2 and 3 belong to different subsets in 7 ways: {{1}{2}{3}{4,5}}, {{1}{2}{5}{3,4}}, {{1}{2}{4}{3,5}}, {{1}{3}{4}{2,5}}, {{1}{3}{5}{2,4}}, {{2}{3}{4}{1,5}} and {{2}{3}{5}{1,4}}.
From _Peter Bala_, Feb 23 2025: (Start)
The array factorizes as
/ 1               \       /1             \ /1             \ /1            \
| 3    1           |     | 3   1          ||0  1           ||0  1          |
| 9    7   1       |  =  | 9   4   1      ||0  3   1       ||0  0  1       | ...
|27   37  12   1   |     |27  13   5  1   ||0  9   4  1    ||0  0  3  1    |
|81  175  97  18  1|     |81  40  18  6  1||0 27  13  5  1 ||0  0  9  4  1 |
|...               |     |...             ||...            ||...           |
where, in the infinite product on the right-hand side, the first array is the Riordan array (1/(1 - 3*x), x/(1 - x)). See A106516. (End)

Peter Bala, Factorising (r,b)-Stirling arrays
Andrei Z. Broder, The r-Stirling numbers, Report Number: CS-TR-82-949, Stanford University, Department of Computer Science; see also, Discrete Math. 49, 241-259 (1984).
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
V. V. Mikhailov, Ordering of some boson operator functions, J. Phys A: Math. Gen. 16 (1983) 3817-3827.
V. V. Mikhailov, Normal ordering and generalised Stirling numbers, J. Phys A: Math. Gen. 18 (1985) 231-235.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001).
L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.
Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.

Cf. A005061 (column 4), A005494 (row sums), A008277, A016753 (column 5), A028025 (column 6), A049458 (matrix inverse), A106516, A143492, A143494, A143496, A143498.

A143495 := (n, k) -> (1/(k-3)!)*add((-1)^(k-i-1)*binomial(k-3,i)*(i+3)^(n-3), i = 0..k-3): for n from 3 to 12 do seq(A143495(n, k), k = 3..n) end do;

nmax = 12; t[n_, k_] := 1/(k-3)!* Sum[ (-1)^(k-j-1)*Binomial[k-3, j]*(j+3)^(n-3), {j, 0, k-3}]; Flatten[ Table[ t[n, k], {n, 3, nmax}, {k, 3, n}]] (* Jean-François Alcover, Dec 07 2011, after Maple *)

@CachedFunction
def stirling2r(n, k, r) :
    if n < r: return 0
    if n == r: return 1 if k == r else 0
    return stirling2r(n-1, k-1, r) + k*stirling2r(n-1, k, r)
A143495 = lambda n, k: stirling2r(n, k, 3)
for n in (3..8): [A143495(n, k) for k in (3..n)] # Peter Luschny, Nov 19 2012

T(n+3,k+3) = (1/k!)*Sum_{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+3)^n, n,k >= 0.
T(n,k) = Stirling2(n,k) - 3*Stirling2(n-1,k) + 2*Stirling2(n-2,k), n,k >= 3.
Recurrence relation: T(n,k) = T(n-1,k-1) + k*T(n-1,k) for n > 3, with boundary conditions: T(n,2) = T(2,n) = 0 for all n; T(3,3) = 1; T(3,k) = 0 for k > 3.
Special cases: T(n,3) = 3^(n-3); T(n,4) = 4^(n-3) - 3^(n-3).
E.g.f. (k+3) column (with offset 3): (1/k!)*exp(3x)*(exp(x)-1)^k.
O.g.f. k-th column: Sum_{n >= k} T(n,k)*x^n = x^k/((1-3*x)*(1-4*x)*...*(1-k*x)).
E.g.f.: exp(3*t + x*(exp(t)-1)) = Sum_{n >= 0} Sum_{k = 0..n} T(n+3,k+3)*x^k*t^n/n! = Sum_{n >= 0} B_n(3;x)*t^n/n! = 1 + (3+x)*t/1! + (9+7*x+x^2)*t^2/2! + ..., where the row polynomials, B_n(3;x) := Sum_{k = 0..n} T(n+3,k+3)*x^k, may be called the 3-Bell polynomials.
Dobinski-type identities: Row polynomial B_n(3;x) = exp(-x)*Sum_{i >= 0} (i+3)^n*x^i/i!; Sum_{k = 0..n} k!*T(n+3,k+3)*x^k = Sum_{i >= 0} (i+3)^n*x^i/(1+x)^(i+1).
The T(n,k) are the connection coefficients between the falling factorials and the shifted monomials (x+3)^(n-3). For example, 9 + 7*x + x*(x-1) = (x+3)^2 and 27 + 37*x + 12x*(x-1) + x*(x-1)*(x-2) = (x+3)^3.
This array is the matrix product P^2 * S, where P denotes Pascal's triangle, A007318 and S denotes the lower triangular array of Stirling numbers of the second kind, A008277 (apply Theorem 10 of [Neuwirth]). The inverse array is A049458, the signed 3-Stirling numbers of the first kind.

A045379 Expansion of e.g.f.: exp(4*z + exp(z) - 1).

Original entry on oeis.org

1, 5, 26, 141, 799, 4736, 29371, 190497, 1291020, 9131275, 67310847, 516369838, 4116416797, 34051164985, 291871399682, 2588914083065, 23733360653955, 224592570163192, 2191466128865567, 22024934452712437, 227771488390279260

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500
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.
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1.

Cf. A000110, A005493, A005494, A108087, A196834.

Equals the row sums of triangle A143496. - Wolfdieter Lang, Sep 29 2011

A045379:= func< n | (&+[Binomial(n,j)*4^(n-j)*Bell(j): j in [0..n]]) >;
[A045379(n): n in [0..30]]; // G. C. Greubel, Dec 01 2022

a[0]= 1; a[n_]:= a[n]= 4*a[n-1] +Sum[Binomial[n-1, k]*a[k], {k,0,n-1}]; Array[a, 21, 0] (* Amiram Eldar, Jul 03 2020 *)

def A045379(n): return sum( 4^(n-j)*bell_number(j)*binomial(n,j) for j in range(n+1))
[A045379(n) for n in range(31)] # G. C. Greubel, Dec 01 2022

a(n) = exp(-1)*Sum_{k>=0} ((k+4)^n)/k!. - Gerald McGarvey, Jun 03 2004
A recursive formula to compute some integer sequences (including A000110, A005493, A005494 and the present sequence). Define G(n, m), where n, m >= 0, as follows: G(0, m) = 1; G(n, m) = G(n-1, m) * (m+1) + G(n-1, m+1), where n > 0. Then G(n, 0) = A000110(n+1); G(n, 1) = A005493(n+1); G(n, 2) = A005494(n+1); G(n, 3) = A045379(n+1). - Alexey Andreev (ava12(AT)nm.ru), Jan 05 2006
Define f_1(x), f_2(x), ... such that f_1(x)=x^3*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,-4). - Milan Janjic, Jul 08 2010
G.f.: 1/U(0) where U(k) = 1 - x*(k+5) - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 4) / LambertW(n)^(n + 9/2). - Vaclav Kotesovec, Jun 10 2020
a(0) = 1; a(n) = 4 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020
a(n) = Sum_{j=0..n} binomial(n, j)*4^(n-j)*A000110(j). - G. C. Greubel, Dec 01 2022

A108087 Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0} (i+k)^n/i!.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 15, 15, 10, 4, 1, 52, 52, 37, 17, 5, 1, 203, 203, 151, 77, 26, 6, 1, 877, 877, 674, 372, 141, 37, 7, 1, 4140, 4140, 3263, 1915, 799, 235, 50, 8, 1, 21147, 21147, 17007, 10481, 4736, 1540, 365, 65, 9, 1, 115975, 115975, 94828, 60814, 29371, 10427, 2727, 537, 82, 10, 1

Offset: 0

Gerald McGarvey, Jun 05 2005

The column for k=0 is A000110 (Bell or exponential numbers). The column for k=1 is A000110 starting at offset 1. The column for k=2 is A005493 (Sum_{k=0..n} k*Stirling2(n,k).). The column for k=3 is A005494 (E.g.f.: exp(3*z+exp(z)-1).). The column for k=4 is A045379 (E.g.f.: exp(4*z+exp(z)-1).). The row for n=0 is 1's sequence, the row for n=1 is the natural numbers. The row for n=2 is A002522 (n^2 + 1.). The row for n=3 is A005491 (n^3 + 3n + 1.). The row for n=4 is A005492.
Number of ways of placing n labeled balls into n+k boxes, where k of the boxes are labeled and the rest are indistinguishable. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006
The column for k = -1 (not shown) is A000296 (Number of partitions of an n-set into blocks of size >1. Also number of cyclically spaced (or feasible) partitions.). - Gerald McGarvey, Oct 08 2006
Equals antidiagonals of an array in which (n+1)-th column is the binomial transform of n-th column, with leftmost column = the Bell sequence, A000110. - Gary W. Adamson, Apr 16 2009
Number of partitions of [n+k] where at least k blocks contain their own index element.  A(2,2) = 10: 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - Alois P. Heinz, Jan 07 2022

			Array A(n,k) begins:
   1,   1,   1,    1,    1,     1,     1,     1,     1,      1, ... A000012;
   1,   2,   3,    4,    5,     6,     7,     8,     9,     10, ... A000027;
   2,   5,  10,   17,   26,    37,    50,    65,    82,    101, ... A002522;
   5,  15,  37,   77,  141,   235,   365,   537,   757,   1031, ... A005491;
  15,  52, 151,  372,  799,  1540,  2727,  4516,  7087,  10644, ... A005492;
  52, 203, 674, 1915, 4736, 10427, 20878, 38699, 67340, 111211, ... ;
Antidiagonal triangle, T(n, k), begins as:
     1;
     1,    1;
     2,    2,    1;
     5,    5,    3,    1;
    15,   15,   10,    4,   1;
    52,   52,   37,   17,   5,   1;
   203,  203,  151,   77,  26,   6,  1;
   877,  877,  674,  372, 141,  37,  7,  1;
  4140, 4140, 3263, 1915, 799, 235, 50,  8,  1;

F. Ruskey, Combinatorial Generation, preprint, 2001.

Alois P. Heinz, Antidiagonals n = 0..140, flattened
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1, Figure 1.
J. Riordan, Letter, Oct 31 1977, The array is on the second page.
F. Ruskey, Combinatorial Generation, 2003.
F. Ruskey, Lexicographic Algorithms [Broken link]

Cf. A000012, A000027, A000110, A002522, A005490, A005491.
Cf. A005492, A005493, A005494, A045379, A086659, A124824.
Main diagonal gives A134980.
Antidiagonal sums give A347420.

A108087:= func< n,k | (&+[Binomial(n-k,j)*k^j*Bell(n-k-j): j in [0..n-k]]) >;
[A108087(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 02 2022

with(combinat):
A:= (n, k)-> add(binomial(n, i) * k^i * bell(n-i), i=0..n):
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jul 18 2012

Unprotect[Power]; 0^0 = 1; A[n_, k_] := Sum[Binomial[n, i] * k^i * BellB[n - i], {i, 0, n}]; Table[Table[A[d - k, k], {k, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *)

f(n,k)=round (suminf(i=0,(i+k)^n/i!)/exp(1));
g(n,k)=for(k=0,k,print1(f(n,k),",")) \\ prints k+1 terms of n-th row

def A108087(n,k): return sum( k^j*bell_number(n-k-j)*binomial(n-k,j) for j in range(n-k+1))
flatten([[A108087(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 02 2022

For n> 1, A(n, k) = k^n + sum_{i=0..n-2} A086659(n, i)*k^i. (A086659 is set partitions of n containing k-1 blocks of length 1, with e.g.f: exp(x*y)*(exp(exp(x)-1-x)-1).)
A(n, k) = k * A(n-1, k) + A(n-1, k+1), A(0, k) = 1. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006
A(n,k) = Sum_{i=0..n} C(n,i) * k^i * Bell(n-i). - Alois P. Heinz, Jul 18 2012
Sum_{k=0..n-1} A(n-k,k) = A005490(n). - Alois P. Heinz, Jan 05 2022
From G. C. Greubel, Dec 02 2022: (Start)
T(n, n) = A000012(n).
T(n, n-1) = A000027(n).
T(n, n-2) = A002522(n-1).
T(n, n-3) = A005491(n-2).
T(n, n-4) = A005492(n+1).
T(2*n, n) = A134980(n).
T(2*n, n+1) = A124824(n), n >= 1.
Sum_{k=0..n} T(n, k) = A347420(n). (End)

A232472 2-Fubini numbers.

Original entry on oeis.org

2, 10, 62, 466, 4142, 42610, 498542, 6541426, 95160302, 1520385010, 26468935022, 498766780786, 10114484622062, 219641848007410, 5085371491003502, 125055112347154546, 3255163896227709422, 89416052656071565810, 2584886208925055791982, 78447137202259689678706, 2493719594804686310662382

Offset: 2

N. J. A. Sloane, Nov 27 2013

			G.f.: 2*x^2 + 10*x^3 + 62*x^4 + 466*x^5 + 4142*x^6 + 42610*x^7 + 498542*x^8 + ...

S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, Unit interval parking functions and the r-Fubini numbers, arXiv:2401.06937 [math.CO], 2024. See page 8.
Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, J. Integer Seq. 17, Article 14.1.1 (2014).

Cf. A000110, A000670, A005493, A005494, A008277, A069321, A143494, A143495, A232472, A232473, A232474.

Column k=1 of A122101.

r:=2; r_Fubini:=func;
[r_Fubini(n, r): n in [r..22]]; // Bruno Berselli, Mar 30 2016

# r-Stirling numbers of second kind (e.g., A008277, A143494, A143495):
T := (n,k,r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r,i)*(i+r)^(n-r),i = 0..k-r):
# r-Bell numbers (e.g. A000110, A005493, A005494):
B := (n,r) -> add(T(n,k,r),k=r..n);
SB := r -> [seq(B(n,r),n=r..30)];
SB(2);
# r-Fubini numbers (e.g., A000670, A232472, A232473, A232474):
F := (n,r) -> add((k)!*T(n,k,r),k=r..n);
SF := r -> [seq(F(n,r),n=r..30)];
SF(2);

Rest[max=20; t=Sum[n^(n - 1) x^n / n!, {n, 1, max}]; 2 Range[0, max]!CoefficientList[Series[D[1 / (1 - y (Exp[x] - 1)), y] /.y->1, {x, 0, max}], x]] (* Vincenzo Librandi Jan 03 2016 *)
Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Table[Fubini[n, 2], {n, 2, 22}] (* Jean-François Alcover, Mar 30 2016 *)

Let A(x) be the g.f. A232472, B(x) the g.f. A000670, then A(x) = (1-x)*B(x) - 1. - Sergei N. Gladkovskii, Nov 29 2013
a(n) = Sum_{k>=2} T_k*k^(n-2)/2^k where T_k is the (k-1)-st triangular number (i.e., T_k = k*(k-1)/2). - Derek Orr, Jan 01 2016
a(n) = 2*A069321(n-1). - Vincenzo Librandi, Jan 03 2016, corrected by Vaclav Kotesovec, Jul 01 2018
a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Jul 01 2018
From Peter Bala, Dec 08 2020: (Start)
a(n+2) = Sum_{k = 0..n} (k+2)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+2)^n ).
a(n+2) = Sum_{k = 0..n} 2^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+2)! ).
a(n) = 2*A069321(n-1) = A000670(n) - A000670(n-1).
a(n+1)= (1/2)*Sum_{k = 0..n} binomial(n,k)*A000670(k+1) for n >= 1.
E.g.f. with offset 0: 2*exp(2*z)/(2 - exp(z))^3 = 2 + 10*z + 62*z^2/2! + 466*z^3/3! + .... (End)

cp's OEIS Frontend

A000178 Superfactorials: product of first n factorials.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

PARI

PARI

Python

Python

Ruby

Formula

A005493 2-Bell numbers: a(n) = number of partitions of [n+1] with a distinguished block.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A056857 Triangle read by rows: T(n,c) = number of successive equalities in set partitions of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

SageMath

Formula

Extensions

A143495 Triangle read by rows: 3-Stirling numbers of the second kind.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A045379 Expansion of e.g.f.: exp(4*z + exp(z) - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula