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A152885
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Number of descents beginning and ending with an odd number in all permutations of {1,2,...,n}.
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2
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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; internal format)
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OFFSET
| 1,3
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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.
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FORMULA
| a(2n)=(2n-1)!*binom(n,2); a(2n+1)=(2n)!*binom(n+1,2).
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EXAMPLE
| a(6)=360 because (i) the descent pairs can be chosen in binom(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.
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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);
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CROSSREFS
| A152886, A152887
Sequence in context: A180982 A195690 A171582 * A052613 A156493 A117515
Adjacent sequences: A152882 A152883 A152884 * A152886 A152887 A152888
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 19 2009
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