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A287899 Number of permutations of [2n] with exactly n cycles such that the elements of each cycle form an integer interval.

%I A287899 #22 Aug 10 2019 09:00:29
%S A287899 1,1,5,31,217,1661,13721,121703,1157857,11826121,129877645,1535504015,
%T A287899 19546846441,267633414517,3932330905361,61806788736551,
%U A287899 1035452546213441,18421374554192017,346790652640704725,6885640002624595007,143771244649798115257
%N A287899 Number of permutations of [2n] with exactly n cycles such that the elements of each cycle form an integer interval.
%C A287899 All terms are odd.
%H A287899 Alois P. Heinz, <a href="/A287899/b287899.txt">Table of n, a(n) for n = 0..448</a>
%H A287899 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A287899 a(n) = A084938(2n,n).
%F A287899 a(n) = [x^n] (1/(1 - x/(1 - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - ...))))))))^n, a continued fraction. - _Ilya Gutkovskiy_, Sep 29 2017
%F A287899 a(n) ~ exp(1) * n * n!. - _Vaclav Kotesovec_, Sep 29 2017
%e A287899 a(2) = 5: (1)(2,3,4), (1)(2,4,3), (1,2)(3,4), (1,2,3)(4), (1,3,2)(4).
%p A287899 b:= proc(n, i) option remember; `if`(n=0 or i=1, n!,
%p A287899        add(b(n-j, i-1)*j!, j=0..n))
%p A287899     end:
%p A287899 a:= n-> b(n$2):
%p A287899 seq(a(n), n=0..25);
%t A287899 Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-Floor[(k + 1)/2]*x, 1, {k, 1, n}])^n, {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Sep 29 2017 *)
%t A287899 Table[SeriesCoefficient[Sum[k!*x^k, {k, 0, n}]^n, {x,0,n}], {n,0,25}] (* _Vaclav Kotesovec_, Aug 10 2019 *)
%Y A287899 Cf. A084938, A088218 (analog for set partitions).
%K A287899 nonn
%O A287899 0,3
%A A287899 _Alois P. Heinz_, Jun 02 2017