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%I #58 Feb 21 2024 08:27:50
%S 1,0,0,1,6,34,216,1566,12840,117696,1193760,13280520,160841520,
%T 2107021680,29689833600,447821503920,7199590366080,122907276334080,
%U 2220524598297600,42328747652446080,849064844592518400,17877531486897734400,394246607165708774400
%N Number of permutations of {1,2,...,n} having no cycles of the form (i, i+1, i+2, ..., i+j-1) (j >= 1).
%C a(n) = A184184(n,0).
%H Seiichi Manyama, <a href="/A184185/b184185.txt">Table of n, a(n) for n = 0..450</a>
%H Patxi Laborde Zubieta, <a href="http://arxiv.org/abs/1505.06098">Occupied corners in tree-like tableaux</a>, arXiv preprint arXiv:1505.06098 [math.CO], 2015.
%F G.f.: (1-z)*F(z-z^2), where F(z) = Sum_{j>=0} j!*z^j (private communication from _Vladeta Jovovic_, May 26 2009).
%F a(n) = Sum_{i=ceiling((n-1)/2)..n} (-1)^(n-i)*i!*binomial(i+1,n-i).
%F G.f.: 1/Q(0), where Q(k) = 1 + x/(1-x) - x*(k+1)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 19 2013
%F a(n) ~ n! / exp(1) * (1 - 1/n - 1/(2*n^2) - 2/(3*n^3) - 23/(24*n^4) - 151/(120*n^5) - 119/(720*n^6) + 14789/(1260*n^7) + 1223843/(13440*n^8) + ...). - _Vaclav Kotesovec_, Nov 30 2021
%F a(n) = (n+2) * a(n-1) - 2 * (n-1) * a(n-2) + (n-2) * a(n-3) for n > 2. - _Seiichi Manyama_, Nov 30 2021
%e a(4)=6 because we have (13)(24), (1432), (1342), (1423), (1243), and (1324).
%p a := proc(n) add((-1)^(n-i)*factorial(i)*binomial(i+1, n-i), i = ceil((1/2)*n-1/2) .. n) end proc: seq(a(n), n = 0 .. 22);
%t a[n_] := Sum[(-1)^(n-i)*i!*Binomial[i+1, n-i], {i, Ceiling[(n-1)/2], n}];
%t Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Nov 29 2017, from Maple *)
%o (PARI) a(n) = sum(k=n\2, n, (-1)^(n-k)*k!*binomial(k+1, n-k)); \\ _Seiichi Manyama_, Nov 30 2021
%o (PARI) a(n) = if(n<3, 0^n, (n+2)*a(n-1)-2*(n-1)*a(n-2)+(n-2)*a(n-3)); \\ _Seiichi Manyama_, Nov 30 2021
%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, k!*x^k*(1-x)^(k+1))) \\ _Seiichi Manyama_, Nov 30 2021
%Y Cf. A013999, A184184.
%K nonn
%O 0,5
%A _Emeric Deutsch_, Feb 16 2011 (based on communication from _Vladeta Jovovic_)
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