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A349788 Number of permutations of [n] having exactly one increasing cycle.

%I A349788 #26 Apr 15 2022 10:32:34
%S A349788 0,1,1,1,5,36,234,1597,12459,111451,1116277,12298958,147655760,
%T A349788 1919465237,26870436345,403044639709,6448695657957,109628096021612,
%U A349788 1973308547820586,37492874766408001,749857477972731979,15747006284752049759,346434131946498886045
%N A349788 Number of permutations of [n] having exactly one increasing cycle.
%C A349788 Cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1) < b(2) < ... .
%C A349788 Exponential convolution of A000587 with A002627.
%H A349788 Alois P. Heinz, <a href="/A349788/b349788.txt">Table of n, a(n) for n = 0..450</a>
%H A349788 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A349788 E.g.f.: exp(1-exp(x))*(exp(x)-1)/(1-x).
%F A349788 a(n) = A186758(n) - A186755(n).
%F A349788 a(n) = Sum_{j=0..n} binomial(n,j)*A000587(j)*A002627(n-j).
%F A349788 a(n) mod 2 = A131719(n).
%F A349788 a(n) ~ (exp(1) - 1) * exp(1 - exp(1)) * n!. - _Vaclav Kotesovec_, Dec 05 2021
%e A349788 a(4) = 5: (1)(243), (143)(2), (142)(3), (132)(4), (1234).
%p A349788 b:= proc(n) option remember; series(`if`(n=0, 1, add((x+
%p A349788      (j-1)!-1)*binomial(n-1, j-1)*b(n-j), j=1..n)), x, 2)
%p A349788     end:
%p A349788 a:= n-> coeff(b(n), x, 1):
%p A349788 seq(a(n), n=0..23);
%t A349788 b[n_] := b[n] = Series[If[n == 0, 1, Sum[(x+
%t A349788      (j-1)!-1)*Binomial[n-1, j-1]*b[n-j], {j, 1, n}]], {x, 0, 2}];
%t A349788 a[n_] := Coefficient[b[n], x, 1];
%t A349788 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Apr 15 2022, after _Alois P. Heinz_ *)
%Y A349788 Column k=1 of A186754.
%Y A349788 Cf. A000587, A002627, A131719, A186755, A186758.
%K A349788 nonn
%O A349788 0,5
%A A349788 _Alois P. Heinz_, Nov 30 2021