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%I #16 Feb 27 2024 15:42:42
%S 1,1,5,70,1960,88200,5821200,529729200,63567504000,9725828112000,
%T 1847907341280000,426866595835680000,117815180450647680000,
%U 38289933646460496000000,14473594918362067488000000,6296013789487499357280000000,3122822839585799681210880000000
%N a(n) is the difference between the number of n-dist-increasing and (n-1)-dist-increasing permutations p of [2n], where p is k-dist-increasing if k>=0 and p(i)<p(i+k) for all i in [2n-k].
%H Alois P. Heinz, <a href="/A370576/b370576.txt">Table of n, a(n) for n = 0..238</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/K-sorted_sequence">K-sorted sequence</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F a(n) = ceiling( (7/9)*(2*n)!/2^n ) = ceiling( (7/9)*A000680(n) ).
%F a(n) = (2*n-1)*n*a(n-1) for n >= 4 with a(0) = a(1) = 1, a(2) = 5, a(3) = 70.
%F a(n) = ceiling( (2n)! * [x^(2n)] (7/9)/(1-x^2/2) ).
%F a(n) = A370505(2n,n).
%e a(2) = 5 = 6 - 1 = |{1234, 1243, 1324, 2134, 2143, 3142}| - |{1234}|.
%p a:= n-> ceil((7/9)*(2*n)!/2^n):
%p seq(a(n), n=0..22);
%p # second Maple program:
%p a:= proc(n) a(n):= `if`(n<4, [1$2, 5, 70][n+1], (2*n-1)*n*a(n-1)) end:
%p seq(a(n), n=0..22);
%Y Cf. A000384, A000680, A370505.
%K nonn,easy
%O 0,3
%A _Alois P. Heinz_, Feb 22 2024
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