A243869 - OEIS

%I #33 Jun 21 2024 15:35:49

%S 1,5,20,70,231,735,2290,7040,21461,65065,196560,592410,1782691,

%T 5358995,16098830,48340180,145107921,435498525,1306845100,3921234350,

%U 11765101151,35298099655,105899891370,317710858920,953154946381,2859509578385,8578618213640

%N Expansion of x^4/[(1+x)*Product_{k=1..3} (1-k*x)].

%C The number of ways to partition a set of n people around a circular table into 4 affinity groups with no two members of a group seated next to each other [Knuth].

%C The first two primes of the sequence are a(5) and a(96). - _Bruno Berselli_, Jun 13 2014

%H G. C. Greubel, <a href="/A243869/b243869.txt">Table of n, a(n) for n = 4..1000</a>

%H J. R. Britnell and M. Wildon, <a href="http://arxiv.org/abs/1507.04803">Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in types A, B and D</a>, arXiv 1507.04803 [math.CO], 2015.

%H D. E. Knuth and O. P. Lossers, <a href="http://www.jstor.org/stable/27642185">Partitions of a circular set</a>, Problem 11151 in Amer. Math. Monthly 114 (3), (2007), p 265, E_4.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,-5,6).

%F a(n) - 3*a(n-1) = A000975(n-3).

%F From _Bruno Berselli_, Jun 13 2014: (Start)

%F G.f.: x^4/(1 - 5*x + 5*x^2 + 5*x^3 - 6*x^4).

%F a(n) = ( 3^n - 4*2^n + (-1)^n + 6 )/24. (End)

%F a(n) = 5*a(n-1) - 5*a(n-2) - 5*a(n-3) + 6*a(n-4). - _Wesley Ivan Hurt_, May 27 2021

%F a(n) = Sum_{i=0..n-1} Stirling2(i,3)*(-1)^(i+n-1). (See _Peter Bala_'s original formula at A105794 dated Jul 10 2013.) - _Igor Victorovich Statsenko_, May 31 2024

%t Table[(3^n - 4 2^n + (-1)^n + 6)/24, {n, 4, 30}] (* _Bruno Berselli_, Jun 13 2014 *)

%o (Magma) [(3^n-4*2^n+(-1)^n+6)/24: n in [4..30]]; // _Bruno Berselli_, Jun 13 2014

%o (PARI) for(n=4,50, print1(( 3^n - 4*2^n + (-1)^n + 6 )/24, ", ")) \\ _G. C. Greubel_, Oct 11 2017

%Y Cf. A000975 (3 affinity groups).

%Y Column k=4 of A261139.

%K nonn,easy

%O 4,2

%A _R. J. Mathar_, Jun 13 2014