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%I #57 Feb 24 2024 10:26:08
%S 0,0,0,0,1,3,9,48,306,2190,17810,162480,1642635,18231465,220420179,
%T 2883693792,40592133316,611765693532,9828843229764,167702100599520,
%U 3028466654021205,57708568527002415,1157199837194069405,24358905149602459920,537053113128448187766
%N Number of permutations of [n] having exactly two adjacent 2-cycles.
%H Seiichi Manyama, <a href="/A370426/b370426.txt">Table of n, a(n) for n = 0..452</a>
%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+2) / (1+x^2)^(k+1).
%F a(n) = (1/2) * Sum_{k=0..floor(n/2)-2} (-1)^k * (n-k-2)! / k!.
%o (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=2, N, k!*x^(k+2)/(1+x^2)^(k+1))/2))
%o (PARI) a(n, k=2, q=2) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;
%Y Column k=2 of A177248.
%Y Cf. A177249, A370524, A370529.
%K nonn
%O 0,6
%A _Seiichi Manyama_, Feb 21 2024
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