A349788 - OEIS

%I #26 Apr 15 2022 10:32:34

%S 0,1,1,1,5,36,234,1597,12459,111451,1116277,12298958,147655760,

%T 1919465237,26870436345,403044639709,6448695657957,109628096021612,

%U 1973308547820586,37492874766408001,749857477972731979,15747006284752049759,346434131946498886045

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

%C 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 Exponential convolution of A000587 with A002627.

%H Alois P. Heinz, <a href="/A349788/b349788.txt">Table of n, a(n) for n = 0..450</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>

%F E.g.f.: exp(1-exp(x))*(exp(x)-1)/(1-x).

%F a(n) = A186758(n) - A186755(n).

%F a(n) = Sum_{j=0..n} binomial(n,j)*A000587(j)*A002627(n-j).

%F a(n) mod 2 = A131719(n).

%F a(n) ~ (exp(1) - 1) * exp(1 - exp(1)) * n!. - _Vaclav Kotesovec_, Dec 05 2021

%e a(4) = 5: (1)(243), (143)(2), (142)(3), (132)(4), (1234).

%p b:= proc(n) option remember; series(`if`(n=0, 1, add((x+

%p      (j-1)!-1)*binomial(n-1, j-1)*b(n-j), j=1..n)), x, 2)

%p     end:

%p a:= n-> coeff(b(n), x, 1):

%p seq(a(n), n=0..23);

%t b[n_] := b[n] = Series[If[n == 0, 1, Sum[(x+

%t      (j-1)!-1)*Binomial[n-1, j-1]*b[n-j], {j, 1, n}]], {x, 0, 2}];

%t a[n_] := Coefficient[b[n], x, 1];

%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Apr 15 2022, after _Alois P. Heinz_ *)

%Y Column k=1 of A186754.

%Y Cf. A000587, A002627, A131719, A186755, A186758.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Nov 30 2021