%I #31 Jan 22 2023 02:37:15

%S 0,1,2,18,72,720,4320,50400,403200,5443200,54432000,838252800,

%T 10059033600,174356582400,2440992153600,47076277248000,

%U 753220435968000,16005934264320000,288106816757760000,6690480522485760000,133809610449715200000,3372002183332823040000

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

%C a(n) is the number of ways to perform the following: Divide the set {1,2,...,n} into three pairwise disjoint subsets, A,B,C so that A union B union C = {1,2,...,n}. Let A contain an odd number of elements and B contain an even number of elements. Linearly order the elements within each subset. - _Geoffrey Critzer_, Sep 26 2011

%D Miklos Bona, A Walk Through Combinatorics, World Scientific Publishing Co., 2002, page 170.

%H Vincenzo Librandi, <a href="/A152887/b152887.txt">Table of n, a(n) for n = 1..400</a>

%F a(2n) = (2n-1)!*C(n+1,2); a(2n+1) = (2n)!*C(n+1,2).

%F E.g.f.: x/((1-x^2)^2*(1-x)). - _Geoffrey Critzer_, Mar 03 2010

%F a(n) = (n-1)!*(2*n*(n+1)+(2*n+1)*(-1)^n-1)/16. - _Bruno Berselli_, Nov 07 2011

%F D-finite with recurrence a(n) -2*a(n-1) +(-n^2+2)*a(n-2) +n*(n-3)*a(n-3)=0. - _R. J. Mathar_, Jul 26 2022

%F Sum_{n>=2} 1/a(n) = 4*(CoshIntegral(1) - gamma - 1/e) + 2 = 4*(A099284 - A001620 - A068985) + 2. - _Amiram Eldar_, Jan 22 2023

%e a(8) = 50400 because (i) the descent pairs can be chosen in 1+2+3+4 = 10 ways, namely (2,1), (4,1), (4,3), (6,1), (6,3), (6,5), (8,1), (8,3), (8,5), (8,7); (ii) they can be placed in 7 positions, namely (1,2), (2,3), (3,4), (4,5), (5,6), (6,7), (7,8); (iii) the remaining 6 entries can be permuted in 6! = 720 ways; 10*7*720 = 50400.

%p a := proc (n) if `mod`(n, 2) = 0 then factorial(n-1)*binomial((1/2)*n+1, 2) else factorial(n-1)*binomial((1/2)*n+1/2, 2) end if end proc: seq(a(n), n = 1 .. 22);

%t CoefficientList[Series[x/((1 - x) (1 - x^2)^2), {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* _Geoffrey Critzer_, Mar 03 2010 *)

%o (Magma) [Factorial(n-1)*(2*n*(n+1)+(2*n+1)*(-1)^n-1)/16: n in [1..20]]; // _Bruno Berselli_, Nov 07 2011

%Y Cf. A152885, A152886.

%Y Cf. A001620, A068985, A099284.

%K nonn

%O 1,3

%A _Emeric Deutsch_, Jan 19 2009