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A152873 Number of permutations of {1,2,...,n} (n>=2) having a single run of even entries. For example, the permutation 513284679 has a single run of even entries: 2846. 1

%I

%S 2,6,12,48,144,720,2880,17280,86400,604800,3628800,29030400,203212800,

%T 1828915200,14631321600,146313216000,1316818944000,14485008384000,

%U 144850083840000,1738201006080000,19120211066880000,248562743869440000

%N Number of permutations of {1,2,...,n} (n>=2) having a single run of even entries. For example, the permutation 513284679 has a single run of even entries: 2846.

%C a(n) = A152667(n,1).

%F a(2n) = (n+1)(n!)^2;

%F a(2n+1) = n!(n+2)!

%F E.g.f.: 24 sqrt(4-x^2)*arcsin(x/2)/[(2-x)^3*(2+x)^2] - x(6-8x-3x^2+2x^3)/ [(2+x)(2-x)^2].

%F G.f.: G(0)/x^2 -1/x^2 -2/x, where G(k) = 1 + x*(k+2)/(1 - x*(k+1)/ (x*(k+1) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 07 2013

%e a(4) = 12 because we have 2413, 2431, 4213, 4231, 1243, 1423 and their reverses.

%p ae := proc (n) options operator, arrow: factorial(n)^2*(n+1) end proc: ao := proc (n) options operator, arrow: factorial(n)*factorial(n+2) end proc: a := proc (n) if `mod`(n, 2) = 0 then ae((1/2)*n) else ao((1/2)*n-1/2) end if end proc; seq(a(n), n = 2 .. 23);

%Y Cf. A152667.

%K nonn

%O 2,1

%A _Emeric Deutsch_, Dec 14 2008

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Last modified June 22 07:09 EDT 2021. Contains 345374 sequences. (Running on oeis4.)