A258213 Number of permutations of {1,2,3,...,n} such that no even numbers are adjacent.

1, 1, 2, 6, 12, 72, 144, 1440, 2880, 43200, 86400, 1814400, 3628800, 101606400, 203212800, 7315660800, 14631321600, 658409472000, 1316818944000, 72425041920000, 144850083840000, 9560105533440000, 19120211066880000, 1491376463216640000, 2982752926433280000

Offset: 0

Author

Ran Pan, May 23 2015

Keywords

• nonn

Crossrefs

Cf. A008279, A010790.

Programs

Maple

a:= n-> (m-> m!^2*(m+1))(iquo(n+1, 2, 'r'))/(2-r):
seq(a(n), n=0..24);  # Alois P. Heinz, Feb 14 2024

PARI

T(n,k) = n!/(n-k)!; \\ A008279
a(n) = ceil(n/2)!*T(ceil(n/2)+1, n\2); \\ Michel Marcus, Nov 24 2022

Formula

a(n) = factorial(ceiling(n/2))*fallfac(ceiling(n/2)+1, floor(n/2)), with fallfac = A008279.

a(2n) = A010790(n), a(2n-1) = A010790(n)/2.

D-finite with recurrence: (4*(n-2)^2 + 24*n - 80)*a(n) + (16*n+24)*a(n-1) - (n+2)*n*((n-2)^2 + 8*n - 17)*a(n-2) = 0. - Georg Fischer, Nov 23 2022