This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370524 #31 Feb 24 2024 14:26:00
%S A370524 0,0,1,2,4,18,99,612,4376,35620,324965,3285270,36462924,440840358,
%T A370524 5767387591,81184266632,1223531387056,19657686459528,335404201199049,
%U A370524 6056933308042410,115417137054004820,2314399674388138810,48717810299204919851,1074106226256896375532
%N A370524 Number of permutations of [n] having exactly one adjacent 2-cycle.
%H A370524 Seiichi Manyama, <a href="/A370524/b370524.txt">Table of n, a(n) for n = 0..450</a>
%H A370524 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 A370524 G.f.: Sum_{k>=1} k! * x^(k+1) / (1+x^2)^(k+1).
%F A370524 a(n) = Sum_{k=0..floor(n/2)-1} (-1)^k * (n-k-1)! / k!.
%e A370524 The permutations of {1,2,3} having exactly one adjacent 2-cycle are (12)(3) and (1)(23). So a(3) = 2.
%o A370524 (PARI) my(N=30, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, k!*x^(k+1)/(1+x^2)^(k+1))))
%o A370524 (PARI) a(n, k=1, q=2) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;
%Y A370524 Column k=2 of A370527.
%Y A370524 Column k=1 of A177248
%Y A370524 Cf. A177249, A370426, A370529.
%K A370524 nonn
%O A370524 0,4
%A A370524 _Seiichi Manyama_, Feb 21 2024