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%I #24 Feb 24 2024 14:25:44
%S 0,0,0,0,1,2,6,24,118,714,5016,40200,362163,3623772,39876540,
%T 478639080,6223394516,87138394540,1307195547720,20916564680760,
%U 355600269756485,6401066270800350,121624180731849810,2432546364331038480,51084540451761077514,1123879093137556106358
%N Number of permutations of [n] having exactly one adjacent 4-cycle.
%H R. A. Brualdi and Emeric Deutsch, <a href="http://arxiv.org/abs/1005.0781">Adjacent q-cycles in permutations</a>, arXiv:1005.0781 [math.CO], 2010.
%F G.f.: Sum_{k>=1} k! * x^(k+3) / (1+x^4)^(k+1).
%F a(n) = Sum_{k=0..floor(n/4)-1} (-1)^k * (n-3*k-3)! / k!.
%e The permutations of {1,2,3,4,5} having exactly one adjacent 4-cycle are (1234)(5) and (1)(2345). So a(5) = 2.
%o (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, k!*x^(k+3)/(1+x^4)^(k+1))))
%o (PARI) a(n, k=1, q=4) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;
%Y Column k=4 of A370527.
%Y Column k=1 of A177252.
%K nonn
%O 0,6
%A _Seiichi Manyama_, Feb 24 2024
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