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A370529 Number of permutations of [n] having exactly three adjacent 2-cycles.

%I A370529 #21 Feb 24 2024 10:24:13
%S A370529 0,0,0,0,0,0,1,4,16,100,730,5940,54160,547540,6077155,73473400,
%T A370529 961231264,13530711096,203921897844,3276281076600,55900700199840,
%U A370529 1009488884673720,19236189509000805,385733279064689820,8119635049867486640,179017704376149395900
%N A370529 Number of permutations of [n] having exactly three adjacent 2-cycles.
%H A370529 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 A370529 G.f.: (1/6) * Sum_{k>=3} k! * x^(k+3) / (1+x^2)^(k+1).
%F A370529 a(n) = (1/6) * Sum_{k=0..floor(n/2)-3} (-1)^k * (n-k-3)! / k!.
%F A370529 a(n) ~ n! / (6*n^3). - _Vaclav Kotesovec_, Feb 21 2024
%p A370529 a:= proc(n) option remember; `if`(n<7, [0$6, 1][n+1], ((n-5)*(n-6)*(n-3)*a(n-1)
%p A370529        -6*(n-4)*a(n-2)+(n-2)*(n-3)*((n-5)*a(n-3)+a(n-4)))/((n-5)*(n-6)))
%p A370529     end:
%p A370529 seq(a(n), n=0..25);  # _Alois P. Heinz_, Feb 21 2024
%o A370529 (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0], Vec(sum(k=3, N, k!*x^(k+3)/(1+x^2)^(k+1))/6))
%o A370529 (PARI) a(n, k=3, q=2) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;
%Y A370529 Column k=3 of A177248.
%Y A370529 Cf. A177249, A370426, A370524.
%K A370529 nonn
%O A370529 0,8
%A A370529 _Seiichi Manyama_, Feb 21 2024