|
%I #18 Dec 04 2022 08:32:48
%S 1,1,1,3,4,20,30,210,336,3024,5040,55440,95040,1235520,2162160,
%T 32432400,57657600,980179200,1764322560,33522128640,60949324800,
%U 1279935820800,2346549004800,53970627110400,99638080819200,2490952020480000,4626053752320000
%N Number of permutations of [n] beginning with at least floor(n/2) ascents.
%H Alois P. Heinz, <a href="/A262033/b262033.txt">Table of n, a(n) for n = 0..732</a>
%F E.g.f.: (x+1)*(exp(x^2)-1)/x^2.
%F a(n) = 2*n*(n*(n-1)*a(n-2)-a(n-1))/((n+2)*(n-1)) for n>1, a(0)=a(1)=1.
%F a(n) = n!/ceiling((n+1)/2)!.
%F a(2n) = A262034(2n) = A001761(n).
%F a(2n+1) = A006963(n+2).
%F Sum_{n>=0} 1/a(n) = 7/4 + 13*exp(1/4)*sqrt(Pi)*erf(1/2)/8, where erf is the error function. - _Amiram Eldar_, Dec 04 2022
%e a(4) = 4: 1234, 1243, 1342, 2341.
%e a(5) = 20: 12345, 12354, 12435, 12453, 12534, 12543, 13425, 13452, 13524, 13542, 14523, 14532, 23415, 23451, 23514, 23541, 24513, 24531, 34512, 34521.
%p a:= proc(n) option remember; `if`(n<2, 1,
%p 2*n*(n*(n-1)*a(n-2)-a(n-1))/((n+2)*(n-1)))
%p end:
%p seq(a(n), n=0..30);
%t a[n_] := n!/Ceiling[(n + 1)/2]!; Array[a, 30, 0] (* _Amiram Eldar_, Dec 04 2022 *)
%Y Cf. A001761, A006963, A262034, A262035.
%K nonn
%O 0,4
%A _Alois P. Heinz_, Sep 08 2015
|
