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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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`%I #9 Jul 31 2022 06:32:46
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`%S 0,1,2,24,96,1080,6480,80640,645120,9072000,90720000,1437004800,
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`%T 17244057600,305124019200,4271736268800,83691159552000,
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`%U 1339058552832000,28810681675776000,518592270163968000,12164510040883200000
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`%N Number of even entries that are followed by a smaller entry in all permutations of {1,2,...,n}.
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`%C a(n) = Sum(k*A134434(n,k), k=0..floor(n/2)).
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`%C 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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`%D S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173-193.
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`%F a(2n) = n(2n)!/2; a(2n+1) = n^2*(2n)!.
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`%F 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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`%e 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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`%p 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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`%Y Cf. A134434, A145890.
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`%K nonn,easy
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`%O 1,3
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`%A _Emeric Deutsch_, Nov 16 2008
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