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A345342 Number of permutations of [2n] having n cycles of the form (c1, c2, ..., c_m) where c1 = min_{i>=1} c_i and c_j = min_{i>=j} c_i or c_j = max_{i>=j} c_i.

%I A345342 #19 Mar 17 2024 07:45:31
%S A345342 1,1,11,195,5033,171465,7264499,368258891,21740278417,1465044247953,
%T A345342 110975742044635,9334724676616339,863320991981279033,
%U A345342 87072657503374176985,9511213780859395685955,1118615909510940858978075,140933163945864346869845025,18937018020284359019138011425
%N A345342 Number of permutations of [2n] having n cycles of the form (c1, c2, ..., c_m) where c1 = min_{i>=1} c_i and c_j = min_{i>=j} c_i or c_j = max_{i>=j} c_i.
%H A345342 Alois P. Heinz, <a href="/A345342/b345342.txt">Table of n, a(n) for n = 0..332</a>
%H A345342 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A345342 a(n) = A344855(2n,n).
%F A345342 a(n) ~ c * (1 + exp(2))^n * (n-1)!, where c = sqrt((exp(2) + 1)/(exp(2) - 1))/(2*Pi) = 0.1823720711148962856100934464088354177502714116352616187167... - _Vaclav Kotesovec_, Jul 15 2021, updated Mar 17 2024
%F A345342 a(0) = 1; a(n) = Sum_{k=0..n} binomial(2*n, n + k + 1)*Stirling2(n + k + 1, k + 1). - _Detlef Meya_, Jan 18 2024
%p A345342 b:= proc(n) option remember; `if`(n=0, 1, add(expand(x*
%p A345342       b(n-j)*binomial(n-1, j-1)*ceil(2^(j-2))), j=1..n))
%p A345342     end:
%p A345342 a:= n-> coeff(b(2*n), x, n):
%p A345342 seq(a(n), n=0..18);
%t A345342 b[n_] := b[n] = If[n == 0, 1, Sum[Expand[x b[n-j] Binomial[n-1, j-1]* Ceiling[2^(j-2)]], {j, n}]];
%t A345342 a[n_] := Coefficient[b[2n], x, n];
%t A345342 Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Aug 25 2021, after _Alois P. Heinz_ *)
%t A345342 a[0] := 1; a[n_] := Sum[Binomial[2*n, n + k + 1]*StirlingS2[n + k + 1, k + 1], {k, 0, n}]; Flatten[Table[a[n] , {n, 0, 17}]] (* _Detlef Meya_, Jan 18 2024 *)
%Y A345342 Cf. A344855.
%K A345342 nonn
%O A345342 0,3
%A A345342 _Alois P. Heinz_, Jun 14 2021