A152886 Number of descents beginning and ending with an even number in all permutations of {1,2,...,n}.
0, 0, 0, 6, 24, 360, 2160, 30240, 241920, 3628800, 36288000, 598752000, 7185024000, 130767436800, 1830744115200, 36614882304000, 585838116864000, 12804747411456000, 230485453406208000, 5474029518397440000, 109480590367948800000, 2810001819444019200000
Offset: 1
Keywords
Examples
a(7) = 2160 because (i) the descent pairs can be chosen in binomial(3,2) = 3 ways, namely (4,2), (6,2), (6,4); (ii) they can be placed in 6 positions, namely (1,2),(2,3),(3,4),(4,5),(5,6),(6,7); (iii) the remaining 5 entries can be permuted in 5! = 120 ways; 3*6*120 = 2160.
Programs
-
Maple
a := proc (n) if `mod`(n, 2) = 0 then factorial(n-1)*binomial((1/2)*n, 2) else factorial(n-1)*binomial((1/2)*n-1/2, 2) end if end proc: seq(a(n), n = 1 .. 22);
-
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,2).
D-finite with recurrence +(-n+4)*a(n) +(n-1)*a(n-1) +(n-2)*(n-1)^2*a(n-2)=0. - R. J. Mathar, Jul 31 2022
Sum_{n>=4} 1/a(n) = 2*(CoshIntegral(1) - gamma - 3*e + 8) = 2*(A099284 - A001620 - 3 * A001113 + 8). - Amiram Eldar, Jan 22 2023