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A130783
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Maximum value of the n-th difference of a permutation of 0..n.
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5
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0, 1, 3, 10, 25, 66, 154, 372, 837, 1930, 4246, 9516, 20618, 45332, 97140, 210664, 447661, 960858, 2028478, 4319100, 9070110, 19188796, 40122028, 84438360, 175913250, 368603716, 765561564, 1598231992, 3310623412, 6889682280, 14238676712, 29551095248
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OFFSET
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0,3
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COMMENTS
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For n>1, a(n) is also the maximum value of the n-th difference of a permutation of 1..n. - Michel Marcus, Apr 15 2017
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LINKS
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F. Disanto, A. Frosini, and S. Rinaldi, Square involutions, J. Int. Seq. 14 (2011) # 11.3.5.
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FORMULA
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a(n) = (n+1)*(2^(n-1)-binomial(n-1,n/2)) if n is even else ((n+1)/2)*(2^n-binomial(n,(n+1)/2)). - Vladeta Jovovic, Aug 23 2007
a(n) = (n+1)*(2^n-binomial(n,[n/2]))/2, where [x] is floor. - Graeme McRae, Jan 30 2012
G.f.: (1-sqrt((1-2*x)/(1+2*x)))/(2*(1-2*x)^2). - Vladeta Jovovic, Aug 24 2007
Asymptotics: a(n) ~ 2^(n-1)*(n+1-sqrt(2*n/Pi)). - Fung Lam, Mar 28 2014
D-finite with recurrence (n-1)*n*a(n) = 2*(n-1)*(n+1)*a(n-1) + 4*(n-2)*n*a(n-2) - 8*(n-1)*n*a(n-3). - Vaclav Kotesovec, Mar 28 2014
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EXAMPLE
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a(1)=1 because 0 1 has a first difference of 1;
a(2)=3 because 2 0 1 has a second difference of 3.
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MAPLE
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MATHEMATICA
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Table[(n + 1) (2^n - Binomial[n, Floor[n/2]])/2, {n, 0, 50}] (* Wesley Ivan Hurt, Nov 25 2013 *)
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PROG
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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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