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A152885 Number of descents beginning and ending with an odd number in all permutations of {1,2,...,n}. 2
0, 0, 2, 6, 72, 360, 4320, 30240, 403200, 3628800, 54432000, 598752000, 10059033600, 130767436800, 2440992153600, 36614882304000, 753220435968000, 12804747411456000, 288106816757760000, 5474029518397440000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

LINKS

Table of n, a(n) for n=1..20.

FORMULA

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

EXAMPLE

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.

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);

CROSSREFS

Cf. A152886, A152887.

Sequence in context: A195690 A329965 A171582 * A295182 A052613 A156493

Adjacent sequences:  A152882 A152883 A152884 * A152886 A152887 A152888

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jan 19 2009

STATUS

approved

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Last modified December 7 00:16 EST 2019. Contains 329812 sequences. (Running on oeis4.)