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 A113583 Number of permutations of length 2*n-1 with no local maxima in even positions. 1

%I

%S 1,4,56,1632,81664,6241280,676506624,98709925888,18655203885056,

%T 4432984678858752,1293646660855398400,454816628946740707328,

%U 189608469405709753122816,92483656403812275277791232,52178449263441077156062429184,33716638014695384983287984291840,24738782851403087736929931445141504

%N Number of permutations of length 2*n-1 with no local maxima in even positions.

%D M. La Croix, A combinatorial proof of a result of Gessel and Greene, Discr. Math., 306 (2006), 2251-2256.

%F E.g.f. (odd powers only): sum(n>=1, a(n) * x^(2*n-1)/(2*n-1)! ) = tanh(x)/(1-x*tanh(x)).

%F a(n)=(2*n-1)!*sum(m=1..2*n, sum(k=0..2*n-2*m, binomial(k+m-1,m-1)*(k+m)!*(-1)^k*2^(2*n-2*m-k)*stirling2(2*n-m,k+m))/(2*n-m)!). [Vladimir Kruchinin, Jun 14 2011]

%o (Maxima)

%o a(n):=(2*n-1)!*sum(sum(binomial(k+m-1,m-1)*(k+m)!*(-1)^k*2^(2*n-2*m-k)*stirling2(2*n-m,k+m),k,0,2*n-2*m)/(2*n-m)!,m,1,2*n); [Vladimir Kruchinin, Jun 14 2011]

%Y Cf. A122647.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Sep 21 2006

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