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

%I A137341 #58 Aug 04 2025 10:14:16
%S A137341 1,1,4,30,360,6240,146160,4420080,166924800,7673823360,420850080000,
%T A137341 27086342976000,2018319704755200,172142484203289600,
%U A137341 16642276683198566400,1808459441303074560000,219273812138054209536000,29473992420094651613184000
%N A137341 a(n) = n! * A000110(n) where A000110 is the sequence of Bell numbers.
%C A137341 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 A137341 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 A137341 W. Lucas and R. Thrall, N-person games in partition function form, Naval Research Logistics Quarterly X, pp. 281-298, 1963.
%H A137341 Vincenzo Librandi, <a href="/A137341/b137341.txt">Table of n, a(n) for n = 0..200</a>
%H A137341 E. T. Bell, <a href="http://www.jstor.org/stable/2300300">Exponential numbers</a>, Amer. Math. Monthly, 41 (1934), 411-419.
%H A137341 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 A137341 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 A137341 a(0) = 1; for n>0, a(n) = Sum_{j=0..n-1} binomial(n-1,j) * a(j) * n!/j!.
%F A137341 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 A137341 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 A137341 a(n) = n! * Sum_{j=0..n} Stirling2(n,j). - _Detlef Meya_, Apr 11 2024
%e A137341 a(0) = 1;
%e A137341 a(1) = C(0,0)*a(0)*1!/0! = 1;
%e A137341 a(2) = C(1,1)*a(1)*2!/1! + C(1,0)*a(0)*2!/0! = 4;
%e A137341 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 A137341 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 A137341 From _Olivier Gérard_, Aug 12 2016: (Start)
%e A137341 Illustration as family of lists of sublists extending set partitions.
%e A137341 In this interpretation the lowercase letters allow us to distinguish between integers introduced at each iteration (or generation).
%e A137341 Construction from the family of size n to family of size n+1 is done by insertion.
%e A137341 Insertion is only possible at the end of a sublist or to create a new singleton sublist at the end of the list.
%e A137341 :
%e A137341 1: {{1a}}*
%e A137341 4: {{1a},{1b}}  {{1a,1b}}  {{1a,2b}}*  {{1a},{2b}}*
%e A137341 30: {{1a,1c},{1b}}     {{1a},{1b,1c}}     {{1a},{1b},{1c}}
%e A137341 ....{{1a,2c},{1b}}     {{1a},{1b,2c}}     {{1a},{1b},{2c}}
%e A137341 ....{{1a,3c},{1b}}     {{1a},{1b,3c}}     {{1a},{1b},{3c}}
%e A137341 ....{{1a,1b,1c}}       {{1a,1b},{1c}}
%e A137341 ....{{1a,1b,2c}}       {{1a,1b},{2c}}
%e A137341 ....{{1a,1b,3c}}       {{1a,1b},{3c}}
%e A137341 ....{{1a,2b,1c}}       {{1a,2b,2c}}       {{1a,2b,3c}}*
%e A137341 ....{{1a,2b},{1c}}     {{1a,2b},{2c}}     {{1a,2b},{3c}}*
%e A137341 ....{{1a,1c},{2b}}     {{1a},{2b,1c}}     {{1a},{2b},{1c}}
%e A137341 ....{{1a,2c},{2b}}     {{1a},{2b,2c}}     {{1a},{2b},{2c}}
%e A137341 ....{{1a,3c},{2b}}*    {{1a},{2b,3c}}*    {{1a},{2b},{3c}}*
%e A137341 :
%e A137341 The lists of sublists marked with * correspond to classical set partitions counted by Bell numbers A000110. (End)
%p A137341 b:= proc(n) option remember; `if`(n=0, 1,
%p A137341       add(b(n-j)*binomial(n-1, j-1), j=1..n))
%p A137341     end:
%p A137341 a:= n-> b(n)*n!:
%p A137341 seq(a(n), n=0..23);  # _Alois P. Heinz_, Aug 30 2019
%t A137341 Table[n!*BellB[n],{n,0,20}] (* _Vaclav Kotesovec_, Mar 13 2014 *)
%o A137341 (Sage) [factorial(m) * bell_number(m) for m in range(17)]  # _Zerinvary Lajos_, Jul 06 2008
%o A137341 (PARI) Vec(serlaplace(serlaplace(exp(exp(O(x^20)+x)-1)))) \\ _Joerg Arndt_, Mar 13 2009
%o A137341 (Python)
%o A137341 from sympy import bell, factorial
%o A137341 [factorial(n) * bell(n) for n in range(101)] # _Indranil Ghosh_, Mar 20 2017
%Y A137341 Cf. A000110, A000142.
%Y A137341 Cf. A048800 = n!*A000110(n-1).
%Y A137341 Main diagonal of A323099 and of A323128.
%K A137341 nonn
%O A137341 0,3
%A A137341 David W. K. Yeung, Eric L. H. Ku and Patricia M. Yeung (wkyeung(AT)hkbu.edu.hk), Apr 08 2008
%E A137341 Edited by _N. J. A. Sloane_, Sep 19 2009
%E A137341 More terms from _Vincenzo Librandi_, Mar 15 2014