A077612 Number of adjacent pairs of form (even,even) among all permutations of {1,2,...,n}.

0, 0, 0, 12, 48, 720, 4320, 60480, 483840, 7257600, 72576000, 1197504000, 14370048000, 261534873600, 3661488230400, 73229764608000, 1171676233728000, 25609494822912000, 460970906812416000, 10948059036794880000, 218961180735897600000, 5620003638888038400000

Offset: 1

Author

Leroy Quet, Frank Ruskey and Dean Hickerson, Nov 11 2002

Keywords

• nonn

Crossrefs

Cf. A000142, A077611, A077613, A110660.

Cf. A001113, A001620, A099284.

Programs

Mathematica

a[n_] := Floor[n/2]*Floor[n/2 - 1]*(n - 1)!; Array[a, 25] (* Amiram Eldar, Jan 22 2023 *)

PARI

a(n) = n\2 * (n\2-1)*(n-1)! ; \\ Michel Marcus, Aug 29 2013

Formula

a(n) = floor(n/2)*floor(n/2-1)*(n-1)!. Proof: There are floor(n/2)*floor(n/2-1) pairs (r, s) with r and s even and distinct. For each pair, there are n-1 places it can occur in a permutation and (n-2)! possible arrangements of the other numbers.

a(n) = A110660(n+2) * A000142(n-1). - Michel Marcus, Aug 29 2013

Sum_{n>=4} 1/a(n) = CoshIntegral(1) - gamma - 3*e + 8 = A099284 - A001620 - 3*A001113 + 8. - Amiram Eldar, Jan 22 2023