cp's OEIS Frontend

A180189 Number of permutations of [n] having exactly 1 circular succession. A circular succession in a permutation p of [n] is either a pair p(i), p(i+1), where p(i+1)=p(i)+1 or the pair p(n), p(1) if p(1)=p(n)+1.

%I A180189 #15 Jul 26 2022 15:01:54
%S A180189 0,2,0,12,40,270,1848,14840,133488,1334970,14684560,176214852,
%T A180189 2290792920,32071101062,481066515720,7697064251760,130850092279648,
%U A180189 2355301661033970,44750731559645088,895014631192902140,18795307255050944520,413496759611120779902
%N A180189 Number of permutations of [n] having exactly 1 circular succession. A circular succession in a permutation p of [n] is either a pair p(i), p(i+1), where p(i+1)=p(i)+1 or the pair p(n), p(1) if p(1)=p(n)+1.
%C A180189 For example, p=(4,1,2,5,3) has 2 circular successions: (1,2) and (3,4).
%H A180189 S. M. Tanny, <a href="http://dx.doi.org/10.1016/0097-3165(76)90063-7">Permutations and successions</a>, J. Combinatorial Theory, Series A, 21 (1976), 196-202.
%F A180189 a(n) = n*(n-1)*d(n-2), where d(j)=A000166(j) are the derangement numbers.
%F A180189 a(n) = A180188(n,1).
%F A180189 E.g.f.: x^2 * exp(-x) / (1 - x). - _Ilya Gutkovskiy_, Oct 11 2021
%F A180189 a(n) = 2 * A000387(n). - _Alois P. Heinz_, Oct 11 2021
%F A180189 D-finite with recurrence (-n+2)*a(n) +n*(n-3)*a(n-1) +n*(n-1)*a(n-2)=0. - _R. J. Mathar_, Jul 26 2022
%e A180189 a(4)=12 because we have 1*243, 142*3, 13*42, 31*24, 3142*, 431*2, 213*4, 4213*, 2*314, 2431*, 42*31, and 3*421 (the circular succession is marked *).
%p A180189 d[0] := 1: for n to 51 do d[n] := n*d[n-1]+(-1)^n end do: seq(n*(n-1)*d[n-2], n = 1 .. 22);
%Y A180189 Cf. A000166, A000387, A180188.
%K A180189 nonn
%O A180189 1,2
%A A180189 _Emeric Deutsch_, Sep 06 2010