A178987 a(n) = n*(n-3)*2^(n-2).

0, -1, -2, 0, 16, 80, 288, 896, 2560, 6912, 17920, 45056, 110592, 266240, 630784, 1474560, 3407872, 7798784, 17694720, 39845888, 89128960, 198180864, 438304768, 964689920, 2113929216, 4613734400, 10032775168, 21743271936, 46976204800, 101200166912

Offset: 0

Author

Paul Curtz, Jan 03 2011

Keywords

• sign
• easy

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), online article 13.6.1.
• Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Crossrefs

Cf. A176027.

Programs

Magma

[n*(n-3)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Aug 04 2011

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 *)

Formula

a(n) = 16*A001793(n-3), n > 3.

a(n) = 8*A001788(n-2)-A052481(n-1). - R. J. Mathar, Jan 04 2011

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