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A077613
Number of adjacent pairs of form (even,odd) among all permutations of {1,2,...,n}. Also, number of adjacent pairs of form (odd,even).
9
0, 1, 4, 24, 144, 1080, 8640, 80640, 806400, 9072000, 108864000, 1437004800, 20118067200, 305124019200, 4881984307200, 83691159552000, 1506440871936000, 28810681675776000, 576213633515520000, 12164510040883200000, 267619220899430400000, 6182004002776842240000
OFFSET
1,3
LINKS
FORMULA
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.
a(n) = A002620(n) * A000142(n-1). - Michel Marcus, Aug 29 2013
Sum_{n>=2} 1/a(n) = 6*(CoshIntegral(1) - gamma) + 2/e - 1 = 6*(A099284 - A001620) + 2*A068985 - 1. - Amiram Eldar, Jan 22 2023
MATHEMATICA
Array[Floor[#/2] Ceiling[#/2] (# - 1)! &, 19] (* Michael De Vlieger, Aug 16 2017 *)
PROG
(PARI) a(n) = floor(n/2)*ceil(n/2)*(n-1)!; \\ Michel Marcus, Aug 29 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved