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%I #38 Sep 08 2022 08:45:15
%S 0,0,1,8,44,208,912,3840,15808,64256,259328,1042432,4180992,16748544,
%T 67047424,268304384,1073463296,4294377472,17178624000,68716855296,
%U 274872401920,1099500093440,4398022393856,17592135712768,70368639320064
%N Half the number of permutations of 0..n with exactly two maxima.
%C Coefficient of the e^(2x) term in the numerator of the n-th derivative of 1/(2-e^x).
%C This sequence, multiplied by 8, appears in a combinatorial problem about DNA chips. - Bruno Petazzoni (bruno(AT)enix.org), Apr 18 2007
%H G. C. Greubel, <a href="/A100575/b100575.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-20,16).
%F From _Paul Barry_, Jan 28 2005: (Start)
%F G.f.: x^2/((1-2*x)^2*(1-4*x)).
%F a(n) = Sum_{k=0..n} (-1)^k*3^(n-k)*binomial(n, k)*floor(k/2). (End)
%F a(n) = 4^(n-1) - (n+1)*2^(n-2). - Bruno Petazzoni (bruno(AT)enix.org), Apr 18 2007
%F a(n+1) = Sum_{k=0..n} k*2^(2*n-1-k). - _Philippe Deléham_ , Oct 29 2013
%F E.g.f.: (1/4)*(exp(4*x) - (1 + 2*x)*exp(2*x)). - _G. C. Greubel_, Mar 21 2022
%e a(2)=1 because there are two maxima in 2,0,1 and 1,0,2
%t d = Drop[ Flatten[ CoefficientList[ Table[ Simplify[ D[1/(2 - E^x), {x, n}]*(E^x - 2)^(n + 1)/E^x], {n, 2, 24}], E^x]], 1]; a = {}; Do[AppendTo[a, Abs[d[[n(n + 1)/2]]]], {n, 23}]; a (* _Robert G. Wilson v_, Dec 01 2004 *)
%t LinearRecurrence[{8,-20,16},{0,0,1},30] (* _Harvey P. Dale_, Apr 21 2020 *)
%o (Magma) [4^(n-1)-(n+1)*2^(n-2): n in [0..30]]; // _Vincenzo Librandi_, Jul 18 2019
%o (Sage) [2^(n-2)*(2^n -(n+1)) for n in (0..30)] # _G. C. Greubel_, Mar 21 2022
%Y Cf. A000431.
%K nonn
%O 0,4
%A _Anthony C Robin_, Nov 29 2004
%E Edited by _Robert G. Wilson v_, Dec 01 2004
%E Definition corrected by Bruno Petazzoni (bruno(AT)enix.org), Apr 13 2007
%E New and simpler definition from _R. H. Hardin_, Aug 09 2007
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