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).

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

Michael De Vlieger, Table of n, a(n) for n = 1..449

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

Cf. A000142, A002620, A077611, A077612.

Cf. A001620, A068985, A099284.

Sequence in context: A067411 A045915 A052609 * A072949 A104531 A225050

Adjacent sequences: A077610 A077611 A077612 * A077614 A077615 A077616

Keyword

nonn

Author

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

Status

approved