%I #30 Mar 15 2021 08:33:48

%S 0,0,0,1,1,3,11,59,369,2665,21823,199983,2028701,22577141,273551115,

%T 3585133147,50540288857,762641865009,12265883397719,209475278413895,

%U 3785852926650453,72191462591370733,1448516763956727331,30507960955933725171,672958104387944656145

%N Number of permutations that have a fixed point and contain 123.

%C A000142(n-2) gives number of permutations with a 123 present.

%C It appears that a(n) = A180191(n-2) - A018934(n-3) for n>3.

%H Alois P. Heinz, <a href="/A193364/b193364.txt">Table of n, a(n) for n = 0..200</a>

%e For n=5 we have 12345, 12354 and 41235, so a(5)=3.

%e For n=6 we have 123456, 123465, 123546, 123465, 123645, 123654, 412356, 451236, 512346, 541236 and 612354, so a(6)=11.

%p a:= proc(n) option remember;

%p `if`(n<7, [0$3, 1$2, 3, 11][n+1],

%p ((4*n^3-42*n^2+92*n+39) *a(n-1)

%p +(32*n^3-2*n^4-163*n^2+223*n+204) *a(n-2)

%p -(n-4)*(n-7)*(2*n^2-10*n-15) *a(n-3)) / (2*n^2-14*n-3))

%p end:

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jan 07 2013

%t a[n_] := a[n] = If[n<7, {0, 0, 0, 1, 1, 3, 11}[[n+1]], ((4n^3 - 42n^2 + 92n + 39) a[n-1] + (32n^3 - 2n^4 - 163n^2 + 223n + 204) a[n-2] - (n-4)(n-7) (2n^2 - 10n - 15) a[n-3])/(2n^2 - 14n - 3)];

%t a /@ Range[0, 30] (* _Jean-François Alcover_, Mar 15 2021, after _Alois P. Heinz_ *)

%Y Cf. A000142, A002467, A018934.

%K nonn

%O 0,6

%A _Jon Perry_, Dec 20 2012