OFFSET
2,2
COMMENTS
There are two ways to change abc: abc -> bca and abc -> cab, that's why we get 2*C(2n,3). There are 2n*(2n-2) = 4n*(n-1) = 8*C(n,2) cases when the two chosen letters are identical, that's why we get -8*C(n,2). Thanks to Miklos Kristof for help.
A diagonal of A059056. - Zerinvary Lajos, Jun 18 2007
With offset "1", a(n) is 16 times the self convolution of n. - Wesley Ivan Hurt, Apr 06 2015
Number of orbits of Aut(Z^7) as function of the infinity norm (n+2) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 53760. - Philippe A.J.G. Chevalier, Dec 28 2015
LINKS
Stefano Spezia, Table of n, a(n) for n = 2..10000
Mark Roger Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, Article P1.32.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = 16*C(n, 3) = 2*C(2*n, 3) - 8*C(n, 2).
From R. J. Mathar, Mar 09 2009: (Start)
G.f.: 16*x^3/(1-x)^4.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).
a(n) = 8*n*(n-1)*(n-2)/3. (End)
a(n) = 16*A000292(n-2). - J. M. Bergot, May 29 2014
E.g.f.: 8*exp(x)*x^3/3. - Stefano Spezia, May 19 2021
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=3} 1/a(n) = 3/32.
Sum_{n>=3} (-1)^(n+1)/a(n) = 3*(8*log(2)-5)/32. (End)
EXAMPLE
a(4) = 64 = 2*C(8,3) - 8*C(4,2) = 2*56 - 8*6 = 112 - 48.
MAPLE
MATHEMATICA
Table[8n(n-1)(n-2)/3, {n, 2, 50}] (* Wesley Ivan Hurt, Apr 06 2015 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 16, 64, 160}, 50] (* Harvey P. Dale, May 20 2021 *)
PROG
(Magma) [8*n*(n-1)*(n-2)/3 : n in [2..50]]; // Wesley Ivan Hurt, Apr 06 2015
(PARI) concat([0], Vec(16*x^3/(1-x)^4+O(x^40))) \\ Stefano Spezia, May 22 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Zerinvary Lajos, Mar 01 2005
STATUS
approved