A077612 - OEIS

%I #19 Jan 22 2023 02:36:40

%S 0,0,0,12,48,720,4320,60480,483840,7257600,72576000,1197504000,

%T 14370048000,261534873600,3661488230400,73229764608000,

%U 1171676233728000,25609494822912000,460970906812416000,10948059036794880000,218961180735897600000,5620003638888038400000

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

%F 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.

%F a(n) = A110660(n+2) * A000142(n-1). - _Michel Marcus_, Aug 29 2013

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

%t a[n_] := Floor[n/2]*Floor[n/2 - 1]*(n - 1)!; Array[a, 25] (* _Amiram Eldar_, Jan 22 2023 *)

%o (PARI) a(n) = n\2 * (n\2-1)*(n-1)! ; \\ _Michel Marcus_, Aug 29 2013

%Y Cf. A000142, A077611, A077613, A110660.

%Y Cf. A001113, A001620, A099284.

%K nonn

%O 1,4

%A _Leroy Quet_, _Frank Ruskey_ and _Dean Hickerson_, Nov 11 2002