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%I #34 Mar 24 2023 11:22:06
%S 0,-1,-2,0,16,80,288,896,2560,6912,17920,45056,110592,266240,630784,
%T 1474560,3407872,7798784,17694720,39845888,89128960,198180864,
%U 438304768,964689920,2113929216,4613734400,10032775168,21743271936
%N a(n) = n*(n-3)*2^(n-2).
%C Binomial transform of 0, -1 followed by A005563.
%C The sequence defines an array by adding higher order differences in successive rows:
%C 0, -1, -2, 0, 16, 80, 288, 896, 2560, 6912, 17920, 45056, 110592, ...
%C -1, -1, 2, 16, 64, 208, 608, 1664, 4352, 11008, 27136, 65536, ... A127276
%C 0, 3, 14, 48, 144, 400, 1056, 2688, 6656, 16128, 38400, 90112, 208896, ... A176027
%C 3, 11, 34, 96, 256, 656, 1632, 3968, 9472, 22272, 51712, 118784, ... A084266
%C 8, 23, 62, 160, 400, 976, 2336, 5504, 12800, 29440, 67072, ...
%C The left column of the array (binomial transform of the sequence) is A067998.
%C For n>2, the sequence gives the number of permutations in the symmetric group S_{n+1} with peaks exactly in positions 2 and n-1. See Theorem 10 in [Billey-Burdzy-Sagan] reference.
%H Vincenzo Librandi, <a href="/A178987/b178987.txt">Table of n, a(n) for n = 0..1000</a>
%H Sara Billey, Krzystof Burdzy and Bruce Sagan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Billey/billey2.html">Permutations With Given Peak Set</a>, J. Integer Sequences, Vol. 16 (2013), online article 13.6.1.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F a(n) = 16*A001793(n-3), n > 3.
%F a(n) = 8*A001788(n-2)-A052481(n-1). - _R. J. Mathar_, Jan 04 2011
%F a(n) = +6*a(n-1) -12*a(n-2) +8*a(n-3).
%F a(n+1)-a(n) = -A127276(n).
%F G.f.: -x*(-1+4*x)/(2*x-1)^3. - _R. J. Mathar_, Jan 04 2011
%F a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k-1) * C(n-1,i). - _Wesley Ivan Hurt_, Sep 20 2017
%t Table[n(n-3)2^(n-2),{n,0,30}] (* or *) LinearRecurrence[{6,-12,8},{0,-1,-2},30] (* _Harvey P. Dale_, Mar 24 2023 *)
%o (Magma) [n*(n-3)*2^(n-2): n in [0..30]]; // _Vincenzo Librandi_, Aug 04 2011
%Y Cf. A176027.
%K sign,easy
%O 0,3
%A _Paul Curtz_, Jan 03 2011
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