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%I #56 Apr 11 2024 17:21:42
%S 1,1,4,30,360,6240,146160,4420080,166924800,7673823360,420850080000,
%T 27086342976000,2018319704755200,172142484203289600,
%U 16642276683198566400,1808459441303074560000,219273812138054209536000,29473992420094651613184000
%N a(n) = n! * A000110(n) where A000110 is the sequence of Bell numbers.
%C 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.
%C 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
%D W. Lucas and R. Thrall, N-person games in partition function form, Naval Research Logistics Quarterly X, pp. 281-298, 1963.
%H Vincenzo Librandi, <a href="/A137341/b137341.txt">Table of n, a(n) for n = 0..200</a>
%H E. T. Bell, <a href="http://www.jstor.org/stable/2300300">Exponential numbers</a>, Amer. Math. Monthly, 41 (1934), 411-419.
%H Abel Lacabanne, <a href="https://arxiv.org/abs/2312.14796">Framization of Schur-Weyl duality and Yokonuma-Hecke type algebras</a>, arXiv:2312.14796 [math.RT], 2023. See p. 34.
%H David W. K. Yeung, Eric L. H. Ku and Patricia M. Yeung, <a href="http://www.m-hikari.com/ija/ija-password-2008/ija-password1-4-2008/yeungIJA1-4-2008.pdf">A Recursive Sequence for the Number of Positioned Partition</a>, International Journal of Algebra, Vol. 2 (2008), No. 4, pp. 181-185.
%F a(0) = 1; for n>0, a(n) = Sum_{j=0..n-1} binomial(n-1,j) * a(j) * n!/j!.
%F 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
%F 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
%F a(n) = n! * Sum_{j, 0, n} Stirling2(n, j). - _Detlef Meya_, Apr 11 2024
%e a(0) = 1;
%e a(1) = C(0,0)*a(0)*1!/0! = 1;
%e a(2) = C(1,1)*a(1)*2!/1! + C(1,0)*a(0)*2!/0! = 4;
%e a(3) = C(2,2)*a(2)*3!/2! + C(2,1)*a(1)*3!/1! + C(2,0)*a(0)*3!/0! = 30;
%e 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.
%e From _Olivier Gérard_, Aug 12 2016: (Start)
%e Illustration as family of lists of sublists extending set partitions.
%e In this interpretation the lowercase letters allow us to distinguish between integers introduced at each iteration (or generation).
%e Construction from the family of size n to family of size n+1 is done by insertion.
%e Insertion is only possible at the end of a sublist or to create a new singleton sublist at the end of the list.
%e :
%e 1: {{1a}}*
%e 4: {{1a},{1b}} {{1a,1b}} {{1a,2b}}* {{1a},{2b}}*
%e 30: {{1a,1c},{1b}} {{1a},{1b,1c}} {{1a},{1b},{1c}}
%e ....{{1a,2c},{1b}} {{1a},{1b,2c}} {{1a},{1b},{2c}}
%e ....{{1a,3c},{1b}} {{1a},{1b,3c}} {{1a},{1b},{3c}}
%e ....{{1a,1b,1c}} {{1a,1b},{1c}}
%e ....{{1a,1b,2c}} {{1a,1b},{2c}}
%e ....{{1a,1b,3c}} {{1a,1b},{3c}}
%e ....{{1a,2b,1c}} {{1a,2b,2c}} {{1a,2b,3c}}*
%e ....{{1a,2b},{1c}} {{1a,2b},{2c}} {{1a,2b},{3c}}*
%e ....{{1a,1c},{2b}} {{1a},{2b,1c}} {{1a},{2b},{1c}}
%e ....{{1a,2c},{2b}} {{1a},{2b,2c}} {{1a},{2b},{2c}}
%e ....{{1a,3c},{2b}}* {{1a},{2b,3c}}* {{1a},{2b},{3c}}*
%e :
%e The lists of sublists marked with * correspond to classical set partitions counted by Bell numbers A000110. (End)
%p b:= proc(n) option remember; `if`(n=0, 1,
%p add(b(n-j)*binomial(n-1, j-1), j=1..n))
%p end:
%p a:= n-> b(n)*n!:
%p seq(a(n), n=0..23); # _Alois P. Heinz_, Aug 30 2019
%t Table[n!*BellB[n],{n,0,20}] (* _Vaclav Kotesovec_, Mar 13 2014 *)
%o (Sage) [factorial(m) * bell_number(m) for m in range(17)] # _Zerinvary Lajos_, Jul 06 2008
%o (PARI) Vec(serlaplace(serlaplace(exp(exp(O(x^20)+x)-1)))) \\ _Joerg Arndt_, Mar 13 2009
%o (Python)
%o from sympy import bell, factorial
%o [factorial(n) * bell(n) for n in range(101)] # _Indranil Ghosh_, Mar 20 2017
%Y Cf. A000110, A000142.
%Y Cf. A048800 = n!*A000110(n-1).
%Y Main diagonal of A323099 and of A323128.
%K nonn,changed
%O 0,3
%A David W. K. Yeung, Eric L. H. Ku and Patricia M. Yeung (wkyeung(AT)hkbu.edu.hk), Apr 08 2008
%E Edited by _N. J. A. Sloane_, Sep 19 2009
%E More terms from _Vincenzo Librandi_, Mar 15 2014
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