This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A047929 #63 Jul 04 2025 18:42:28
%S A047929 0,18,96,300,720,1470,2688,4536,7200,10890,15840,22308,30576,40950,
%T A047929 53760,69360,88128,110466,136800,167580,203280,244398,291456,345000,
%U A047929 405600,473850,550368,635796,730800,836070,952320,1080288,1220736
%N A047929 a(n) = n^2*(n-1)*(n-2).
%C A047929 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
%C A047929 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 MartÃnez_, Feb 24 2014
%H A047929 Vincenzo Librandi, <a href="/A047929/b047929.txt">Table of n, a(n) for n = 2..1000</a>
%H A047929 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H A047929 Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%H A047929 <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>
%H A047929 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A047929 a(n) = A004320(n-2)*6.
%F A047929 G.f.: 6*x^3*(3 + x)/(1 - x)^5. - _Stefano Spezia_, May 20 2021
%F A047929 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
%F A047929 From _Amiram Eldar_, May 25 2021: (Start)
%F A047929 Sum_{n>=3} 1/a(n) = (Pi^2 - 9)/12.
%F A047929 Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/24 + 2*log(2) - 7/4. (End)
%t A047929 Drop[CoefficientList[Series[6 x^3*(3 + x)/(1 - x)^5, {x, 0, 34}], x], 2] (* _Michael De Vlieger_, May 21 2021 *)
%o A047929 (Magma) [n^2*(n-1)*(n-2): n in [2..40]]; // _Vincenzo Librandi_, May 02 2011
%o A047929 (PARI) a(n)=n^4 - 3*n^3 + 2*n^2 \\ _Charles R Greathouse IV_, May 02 2011
%Y A047929 Cf. A004320, A045991, A233440.
%K A047929 nonn,easy
%O A047929 2,2
%A A047929 _N. J. A. Sloane_