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A137341 a(n) = n!*A000110(n) where A000110 is the sequence of Bell numbers. 6
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)
OFFSET

0,3

COMMENTS

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

REFERENCES

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

LINKS

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.

FORMULA

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

EXAMPLE

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)

MATHEMATICA

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

PROG

(Sage) [factorial(m)*bell_number(m) for m in xrange (0, 17)] # Zerinvary Lajos, Jul 06 2008

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

(Python)

from sympy import bell, factorial

print [factorial(n) * bell(n) for n in xrange(0, 101)] # Indranil Ghosh, Mar 20 2017

CROSSREFS

Cf. A000110.

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

Sequence in context: A218296 A118792 A192549 * A166892 A185523 A187736

Adjacent sequences:  A137338 A137339 A137340 * A137342 A137343 A137344

KEYWORD

nonn,changed

AUTHOR

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

EXTENSIONS

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

More terms from Vincenzo Librandi, Mar 15 2014

STATUS

approved

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Last modified March 28 11:45 EDT 2017. Contains 284186 sequences.