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%I #26 Jan 22 2023 02:36:48
%S 0,1,4,24,144,1080,8640,80640,806400,9072000,108864000,1437004800,
%T 20118067200,305124019200,4881984307200,83691159552000,
%U 1506440871936000,28810681675776000,576213633515520000,12164510040883200000,267619220899430400000,6182004002776842240000
%N Number of adjacent pairs of form (even,odd) among all permutations of {1,2,...,n}. Also, number of adjacent pairs of form (odd,even).
%H Michael De Vlieger, <a href="/A077613/b077613.txt">Table of n, a(n) for n = 1..449</a>
%F a(n) = floor(n/2)*ceiling(n/2)*(n-1)!. Proof: There are floor(n/2)*ceiling(n/2) pairs (r, s) with r even and s odd. 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) = A002620(n) * A000142(n-1). - _Michel Marcus_, Aug 29 2013
%F Sum_{n>=2} 1/a(n) = 6*(CoshIntegral(1) - gamma) + 2/e - 1 = 6*(A099284 - A001620) + 2*A068985 - 1. - _Amiram Eldar_, Jan 22 2023
%t Array[Floor[#/2] Ceiling[#/2] (# - 1)! &, 19] (* _Michael De Vlieger_, Aug 16 2017 *)
%o (PARI) a(n) = floor(n/2)*ceil(n/2)*(n-1)!; \\ _Michel Marcus_, Aug 29 2013
%Y Cf. A000142, A002620, A077611, A077612.
%Y Cf. A001620, A068985, A099284.
%K nonn
%O 1,3
%A _Leroy Quet_, _Frank Ruskey_ and _Dean Hickerson_, Nov 11 2002
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