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Number of set partitions of [n] into exactly five parts such that no part contains two elements with a circular distance less than three.
2

%I #9 Apr 16 2023 21:53:49

%S 1,3,14,44,138,458,1397,4219,12974,39270,118004,355924,1071425,

%T 3217011,9662070,29016280,87083486,261322358,784185253,2352907211,

%U 7059288006,21179391714,63541238088,190628647440,571896993217,1715715461187,5147189398142,15441653075684

%N Number of set partitions of [n] into exactly five parts such that no part contains two elements with a circular distance less than three.

%H Alois P. Heinz, <a href="/A261481/b261481.txt">Table of n, a(n) for n = 5..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,12,-6,-18,-47,-32,17,36,36).

%F G.f.: -(6*x^5+6*x^4+x^3+7*x^2+x+1)*x^5 / ((x-1) *(3*x-1) *(2*x-1) *(x+1) *(x^2+x+1) *(3*x^2+x+1) *(2*x^2+x+1)).

%F a(n) = 2*a(n-1) + a(n-2) + 12*a(n-3) - 6*a(n-4) - 18*a(n-5) - 47*a(n-6) - 32*a(n-7) + 17*a(n-8) + 36*a(n-9) + 36*a(n-10). - _Wesley Ivan Hurt_, Apr 16 2023

%e a(5) = 1: 1|2|3|4|5.

%e a(6) = 3: 14|2|3|5|6, 1|25|3|4|6, 1|2|36|4|5.

%e a(7) = 14: 14|25|3|6|7, 14|26|3|5|7, 14|2|36|5|7, 14|2|37|5|6, 15|26|3|4|7, 15|2|36|4|7, 15|2|37|4|6, 15|2|3|47|6, 1|25|36|4|7, 1|25|37|4|6, 1|25|3|47|6, 1|26|37|4|5, 1|26|3|47|5, 1|2|36|47|5.

%Y Column k=5 of A261477.

%K nonn,easy

%O 5,2

%A _Alois P. Heinz_, Aug 20 2015