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%I #15 Feb 24 2024 10:49:54
%S 0,0,0,0,0,0,0,0,1,3,12,60,357,2508,20100,181080,1811886,19938270,
%T 239319540,3111697260,43569197270,653597773860,10458282340380,
%U 177800134878240,3200533135400175,60812090365924905,1216273182165519240,25542270225880538760
%N Number of permutations of [n] having exactly two adjacent 4-cycles.
%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.: (1/2) * Sum_{k>=2} k! * x^(k+6) / (1+x^4)^(k+1).
%F a(n) = (1/2) * Sum_{k=0..floor(n/4)-2} (-1)^k * (n-3*k-6)! / k!.
%o (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=2, N, k!*x^(k+6)/(1+x^4)^(k+1))/2))
%o (PARI) a(n, k=2, q=4) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;
%Y Column k=2 of A177252.
%Y Cf. A370426, A370528.
%K nonn
%O 0,10
%A _Seiichi Manyama_, Feb 24 2024