A370576 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)

1, 1, 5, 70, 1960, 88200, 5821200, 529729200, 63567504000, 9725828112000, 1847907341280000, 426866595835680000, 117815180450647680000, 38289933646460496000000, 14473594918362067488000000, 6296013789487499357280000000, 3122822839585799681210880000000

Offset: 0

Author

Alois P. Heinz, Feb 22 2024

Keywords

• nonn
• easy

Examples

			a(2) = 5 = 6 - 1 = |{1234, 1243, 1324, 2134, 2143, 3142}| - |{1234}|.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..238
• Wikipedia, K-sorted sequence
• Wikipedia, Permutation

Crossrefs

Cf. A000384, A000680, A370505.

Programs

Maple

a:= n-> ceil((7/9)*(2*n)!/2^n):
seq(a(n), n=0..22);
# second Maple program:
a:= proc(n) a(n):= `if`(n<4, [1$2, 5, 70][n+1], (2*n-1)*n*a(n-1)) end:
seq(a(n), n=0..22);

Formula

a(n) = ceiling( (7/9)*(2*n)!/2^n ) = ceiling( (7/9)*A000680(n) ).

a(n) = (2*n-1)*n*a(n-1) for n >= 4 with a(0) = a(1) = 1, a(2) = 5, a(3) = 70.

a(n) = ceiling( (2n)! * [x^(2n)] (7/9)/(1-x^2/2) ).

a(n) = A370505(2n,n).