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A137341 a(n) = n! * A000110(n) where A000110 is the sequence of Bell numbers. 12
1, 1, 4, 30, 360, 6240, 146160, 4420080, 166924800, 7673823360, 420850080000, 27086342976000, 2018319704755200, 172142484203289600, 16642276683198566400, 1808459441303074560000, 219273812138054209536000, 29473992420094651613184000 (list; graph; refs; listen; history; text; internal format)



Cooperative games are frequently formulated in terms of partition functions. In particular, the set of players may be divided into various coalitions forming partitions with different coalition structures. This recursive sequence identifies the number of partitions in an n-player game where the position of the individual player counts.

Lists of sublists of total size n with up to n different 1s, up to n-1 different 2s, ... generated by successive insertion. Sublists stay ordered as inserted. See example field for illustration. - Olivier Gérard, Aug 12 2016


W. Lucas and R. Thrall, N-person games in partition function form, Naval Research Logistics Quarterly X, pp. 281-298, 1963.


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

E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.

David W. K. Yeung, Eric L. H. Ku and Patricia M. Yeung, A Recursive Sequence for the Number of Positioned Partition, International Journal of Algebra, Vol. 2 (2008), No. 4, pp. 181-185.


a(0) = 1; for n>0, a(n) = Sum_{j=0..n-1} binomial(n-1,j) * a(j) * n!/j!.

From the recurrence it follows that A'(x) = exp(x) * A(x) where A(x) = Sum_{k>=0} a(k) * x^k / k!^2. The solution to this differential equation is A(x) = exp(exp(x)). The expression in Joerg Arndt's PARI program follows from this. - Max Alekseyev, Mar 11 2009

a(n) ~ n^(2*n+1/2) * exp(n/LambertW(n)-1-2*n) * sqrt(2*Pi/(1+LambertW(n))) / LambertW(n)^n. - Vaclav Kotesovec, Mar 13 2014


a(0) = 1;

a(1) = C(0,0)*a(0)*1!/0! = 1;

a(2) = C(1,1)*a(1)*2!/1! + C(1,0)*a(0)*2!/0! = 4;

a(3) = C(2,2)*a(2)*3!/2! + C(2,1)*a(1)*3!/1! + C(2,0)*a(0)*3!/0! = 30;

a(4) = C(3,3)*a(3)*4!/3! + C(3,2)*a(2)*4!/2! + C(3,1)*a(1)*4!/1! + C(3,0)*a(0)*4!/0! = 360.

From Olivier Gérard, Aug 12 2016: (Start)

Illustration as family of lists of sublists extending set partitions.

In this interpretation the lowercase letters allow us to distinguish between integers introduced at each iteration (or generation).

Construction from the family of size n to family of size n+1 is done by insertion.

Insertion is only possible at the end of a sublist or to create a new singleton sublist at the end of the list.


1: {{1a}}*

4: {{1a},{1b}}  {{1a,1b}}  {{1a,2b}}*  {{1a},{2b}}*

30: {{1a,1c},{1b}}     {{1a},{1b,1c}}     {{1a},{1b},{1c}}

....{{1a,2c},{1b}}     {{1a},{1b,2c}}     {{1a},{1b},{2c}}

....{{1a,3c},{1b}}     {{1a},{1b,3c}}     {{1a},{1b},{3c}}

....{{1a,1b,1c}}       {{1a,1b},{1c}}

....{{1a,1b,2c}}       {{1a,1b},{2c}}

....{{1a,1b,3c}}       {{1a,1b},{3c}}

....{{1a,2b,1c}}       {{1a,2b,2c}}       {{1a,2b,3c}}*

....{{1a,2b},{1c}}     {{1a,2b},{2c}}     {{1a,2b},{3c}}*

....{{1a,1c},{2b}}     {{1a},{2b,1c}}     {{1a},{2b},{1c}}

....{{1a,2c},{2b}}     {{1a},{2b,2c}}     {{1a},{2b},{2c}}

....{{1a,3c},{2b}}*    {{1a},{2b,3c}}*    {{1a},{2b},{3c}}*


The lists of sublists marked with * correspond to classical set partitions counted by Bell numbers A000110. (End)


b:= proc(n) option remember; `if`(n=0, 1,

      add(b(n-j)*binomial(n-1, j-1), j=1..n))


a:= n-> b(n)*n!:

seq(a(n), n=0..23);  # Alois P. Heinz, Aug 30 2019


Table[n!*BellB[n], {n, 0, 20}] (* Vaclav Kotesovec, Mar 13 2014 *)


(Sage) [factorial(m) * bell_number(m) for m in range(17)]  # Zerinvary Lajos, Jul 06 2008

(PARI) Vec(serlaplace(serlaplace(exp(exp(O(x^20)+x)-1)))) \\ Joerg Arndt, Mar 13 2009


from sympy import bell, factorial

[factorial(n) * bell(n) for n in range(101)] # Indranil Ghosh, Mar 20 2017


Cf. A000110, A000142.

Cf. A048800 = n!*A000110(n-1).

Main diagonal of A323099 and of A323128.

Sequence in context: A317030 A192549 A303001 * A295899 A291060 A166892

Adjacent sequences:  A137338 A137339 A137340 * A137342 A137343 A137344




David W. K. Yeung, Eric L. H. Ku and Patricia M. Yeung (wkyeung(AT)hkbu.edu.hk), Apr 08 2008


Edited by N. J. A. Sloane, Sep 19 2009

More terms from Vincenzo Librandi, Mar 15 2014



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Last modified October 6 02:32 EDT 2022. Contains 357261 sequences. (Running on oeis4.)