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A145889
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Number of even entries that are followed by a smaller entry in all permutations of {1,2,...,n}.
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0
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0, 1, 2, 24, 96, 1080, 6480, 80640, 645120, 9072000, 90720000, 1437004800, 17244057600, 305124019200, 4271736268800, 83691159552000, 1339058552832000, 28810681675776000, 518592270163968000, 12164510040883200000
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OFFSET
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1,3
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COMMENTS
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a(n) = Sum(k*A134434(n,k), k=0..floor(n/2)).
The average of the number of even entries that start a descent over all permutations of {1,2,...n} is (1/n)[floor(n/2)]^2.
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REFERENCES
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S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173-193.
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LINKS
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FORMULA
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a(2n) = n(2n)!/2; a(2n+1) = n^2*(2n)!.
D-finite with recurrence (-4*n+11)*a(n) +(9*n-25)*a(n-1) +(n-2)*(4*n^2-3*n-3)*a(n-2) -(n-2)*(n-3)*(5*n-7)*a(n-3)=0. - R. J. Mathar, Jul 31 2022
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EXAMPLE
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a(3)=2 because the permutations of {1,2,3} are 123, 132, 2'13, 231, 312 and 32'1 with the even entries that start a descent marked.
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MAPLE
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a:=proc(n) if `mod`(n, 2)=0 then (1/4)*n*factorial(n) else (1/4)*(n-1)^2*factorial(n-1) end if end proc: seq(a(n), n=1..20);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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