A185652 - OEIS

%I #17 Oct 26 2021 14:28:19

%S 0,0,1,0,5,18,89,519,3853,27555,233431,2167152,21596120,232817282,

%T 2718706924,33814848445,448311181346,6319365554730,94225534689624,

%U 1481940898130323,24536143182460549,426432943716156580,7762187693343502658,147704506384475066381

%N Number of permutations of [n] having a shortest ascending run of length 2.

%H Alois P. Heinz, <a href="/A185652/b185652.txt">Table of n, a(n) for n = 0..150</a>

%F a(n) ~ c * (3*sqrt(3)/(2*Pi))^n * n!, where c = 0.45178068752734823... . - _Vaclav Kotesovec_, Sep 06 2014

%e a(2) = 1: 12.

%e a(4) = 5: 1324, 1423, 2314, 2413, 3412.

%e a(5) = 18: 12435, 12534, 13245, 13425, 13524, 14235, 14523, 15234, 23145, 23415, 23514, 24135, 24513, 25134, 34125, 34512, 35124, 45123.

%t A[n_, k_] := A[n, k] = Module[{b}, b[u_, o_, t_] := b[u, o, t] = If[t + o <= k, (u + o)!, Sum[b[u + i - 1, o - i, Min[k, t] + 1], {i, 1, o}] + If[t <= k, u (u + o - 1)!, Sum[b[u - i, o + i - 1, 1], {i, 1, u}]]]; Sum[b[j - 1, n - j, 1], {j, 1, n}]];

%t a[n_] := A[n, 2] - A[n, 1];

%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Oct 26 2021, after _Alois P. Heinz_ in A064315 *)

%Y Column k=2 of A064315.

%Y Cf. A086089 (3*sqrt(3)/(2*Pi)).

%K nonn

%O 0,5

%A _Alois P. Heinz_, Aug 29 2013