A152885 Number of descents beginning and ending with an odd number in all permutations of {1,2,...,n}.

0, 0, 2, 6, 72, 360, 4320, 30240, 403200, 3628800, 54432000, 598752000, 10059033600, 130767436800, 2440992153600, 36614882304000, 753220435968000, 12804747411456000, 288106816757760000, 5474029518397440000, 133809610449715200000, 2810001819444019200000

Offset: 1

Author

Emeric Deutsch, Jan 19 2009

Keywords

• nonn

Comments

a(n) is also number of descents beginning with an odd number and ending with an even number in all permutations of {1,2,...,n}. Example: a(4)=6; indeed for n=4 the only descent to be counted is 32, occurring only in 1324, 1432, 4132, 3214, 3241 and 4321.

Examples

			a(6) = 360 because (i) the descent pairs can be chosen in binomial(3,2) = 3 ways, namely (3,1), (5,1), (5,3); (ii) they can be placed in 5 positions, namely (1,2),(2,3),(3,4),(4,5),(5,6); (iii) the remaining 4 entries can be permuted in 4!=24 ways; 3*5*24 = 360.
		

Crossrefs

Cf. A152886, A152887.

Cf. A001620, A073742, A099284.

Programs

Maple

a := proc (n) if `mod`(n, 2) = 0 then (1/4)*factorial(n)*((1/2)*n-1) else (1/8)*(n-1)*(n+1)*factorial(n-1) end if end proc: seq(a(n), n = 1 .. 20);

Mathematica

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

Formula

a(2n) = (2n-1)!*binomial(n,2); a(2n+1) = (2n)!*binomial(n+1,2).

D-finite with recurrence (-n+3)*a(n) +(n-1)*a(n-1) +n*(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Jul 26 2022

Sum_{n>=3} 1/a(n) = 8*(CoshIntegral(1) - gamma - sinh(1) + 1) = 8*(A099284 - A001620 - A073742 + 1). - Amiram Eldar, Jan 22 2023