A243869 Expansion of x^4/[(1+x)*Product_{k=1..3} (1-k*x)].

1, 5, 20, 70, 231, 735, 2290, 7040, 21461, 65065, 196560, 592410, 1782691, 5358995, 16098830, 48340180, 145107921, 435498525, 1306845100, 3921234350, 11765101151, 35298099655, 105899891370, 317710858920, 953154946381, 2859509578385, 8578618213640

Offset: 4

Author

R. J. Mathar, Jun 13 2014

Keywords

• nonn
• easy

Comments

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].

The first two primes of the sequence are a(5) and a(96). - Bruno Berselli, Jun 13 2014

Links

• G. C. Greubel, Table of n, a(n) for n = 4..1000
• J. R. Britnell and M. Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in types A, B and D, arXiv 1507.04803 [math.CO], 2015.
• D. E. Knuth and O. P. Lossers, Partitions of a circular set, Problem 11151 in Amer. Math. Monthly 114 (3), (2007), p 265, E_4.
• Index entries for linear recurrences with constant coefficients, signature (5,-5,-5,6).

Crossrefs

Cf. A000975 (3 affinity groups).

Column k=4 of A261139.

Programs

Magma

[(3^n-4*2^n+(-1)^n+6)/24: n in [4..30]]; // Bruno Berselli, Jun 13 2014

Mathematica

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

PARI

for(n=4,50, print1(( 3^n - 4*2^n + (-1)^n + 6 )/24, ", ")) \\ G. C. Greubel, Oct 11 2017

Formula

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

From Bruno Berselli, Jun 13 2014: (Start)

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

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

a(n) = 5*a(n-1) - 5*a(n-2) - 5*a(n-3) + 6*a(n-4). - Wesley Ivan Hurt, May 27 2021

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