OFFSET
0,3
COMMENTS
Binomial transform of 0, -1 followed by A005563.
The sequence defines an array by adding higher order differences in successive rows:
0, -1, -2, 0, 16, 80, 288, 896, 2560, 6912, 17920, 45056, 110592, ...
-1, -1, 2, 16, 64, 208, 608, 1664, 4352, 11008, 27136, 65536, ... A127276
0, 3, 14, 48, 144, 400, 1056, 2688, 6656, 16128, 38400, 90112, 208896, ... A176027
3, 11, 34, 96, 256, 656, 1632, 3968, 9472, 22272, 51712, 118784, ... A084266
8, 23, 62, 160, 400, 976, 2336, 5504, 12800, 29440, 67072, ...
The left column of the array (binomial transform of the sequence) is A067998.
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.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Sara Billey, Krzystof Burdzy and Bruce Sagan, Permutations With Given Peak Set, J. Integer Sequences, Vol. 16 (2013), Article 13.6.1.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).
FORMULA
a(n) = 16*A001793(n-3), n > 3.
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3).
a(n+1)-a(n) = -A127276(n).
G.f.: -x*(-1+4*x)/(2*x-1)^3. - R. J. Mathar, Jan 04 2011
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k-1) * C(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
a(n) = Sum_{k=0..n} k^2 * (-1)^k * 3^(n-k) * binomial(n,k). - Seiichi Manyama, Apr 18 2025
From Amiram Eldar, Dec 30 2025: (Start)
Sum_{n>=4} 1/a(n) = 8/9 - 7*log(2)/6.
Sum_{n>=4} (-1)^n/a(n) = (3/2)*log(3/2) - 5/9. (End)
E.g.f.: x*exp(2*x)*(x - 1). - Enrique Navarrete, Feb 19 2026
MATHEMATICA
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 *)
PROG
(Magma) [n*(n-3)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Aug 04 2011
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Paul Curtz, Jan 03 2011
STATUS
approved