A047929 a(n) = n^2*(n-1)*(n-2).

0, 18, 96, 300, 720, 1470, 2688, 4536, 7200, 10890, 15840, 22308, 30576, 40950, 53760, 69360, 88128, 110466, 136800, 167580, 203280, 244398, 291456, 345000, 405600, 473850, 550368, 635796, 730800, 836070, 952320, 1080288, 1220736

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

There are 5 ways to put parentheses in the expression a - b - c - d: (a - (b - c)) - d, ((a - b) - c) - d, (a - b) - (c - d), a - (b - (c - d)), a - ((b - c) - d). This sequence describes how many sets of natural numbers [a,b,c,d] can be produced with the numbers {0,1,2,3,...,n} such that all the distinct expressions take different values. A045991 describes the similar process for a - b - c. For example, no sets can be produced with only 0's or only 0's and 1's; with {0,1,2,3}, 18 such sets can be produced. - Asher Auel, Jan 26 2000

For n >= 3, a(n)/6 is the number of permutations of n symbols that 3-commute with an n-cycle (see A233440 for definition). - Luis Manuel Rivera Martínez, Feb 24 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..1000
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
• Index entries for sequences related to parenthesizing
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A004320, A045991, A233440.

Programs

Magma

[n^2*(n-1)*(n-2): n in [2..40]]; // Vincenzo Librandi, May 02 2011

Mathematica

Drop[CoefficientList[Series[6 x^3*(3 + x)/(1 - x)^5, {x, 0, 34}], x], 2] (* Michael De Vlieger, May 21 2021 *)

PARI

a(n)=n^4 - 3*n^3 + 2*n^2 \\ Charles R Greathouse IV, May 02 2011

Formula

a(n) = A004320(n-2)*6.

G.f.: 6*x^3*(3 + x)/(1 - x)^5. - Stefano Spezia, May 20 2021

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, May 22 2021

From Amiram Eldar, May 25 2021: (Start)

Sum_{n>=3} 1/a(n) = (Pi^2 - 9)/12.

Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/24 + 2*log(2) - 7/4. (End)