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 A243869 Expansion of x^4/[(1+x)*Product_{k=1..3} (1-k*x)]. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 4,2 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, M. Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in types A, B and D, arXiv 1507.04803, 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). 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 MATHEMATICA Table[(3^n - 4 2^n + (-1)^n + 6)/24, {n, 4, 30}] (* Bruno Berselli, Jun 13 2014 *) PROG (MAGMA) [(3^n-4*2^n+(-1)^n+6)/24: n in [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 CROSSREFS Cf. A000975 (3 affinity groups). Column k=4 of A261139. Sequence in context: A000343 A005324 A304011 * A154638 A054889 A056384 Adjacent sequences:  A243866 A243867 A243868 * A243870 A243871 A243872 KEYWORD nonn,easy AUTHOR R. J. Mathar, Jun 13 2014 STATUS approved

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Last modified August 4 15:53 EDT 2021. Contains 346447 sequences. (Running on oeis4.)