A102860 Number of ways to change three non-identical letters in the word aabbccdd..., where there are n types of letters.

0, 16, 64, 160, 320, 560, 896, 1344, 1920, 2640, 3520, 4576, 5824, 7280, 8960, 10880, 13056, 15504, 18240, 21280, 24640, 28336, 32384, 36800, 41600, 46800, 52416, 58464, 64960, 71920, 79360, 87296, 95744, 104720, 114240, 124320, 134976, 146224

Offset: 2

Author

Zerinvary Lajos, Mar 01 2005

Keywords

• easy
• nonn

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

Examples

			a(4) = 64 = 2*C(8,3) - 8*C(4,2) = 2*56 - 8*6 = 112 - 48.
		

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).

Crossrefs

Cf. A000292, A046092, A059056.

Programs

Magma

[8*n*(n-1)*(n-2)/3 : n in [2..50]]; // Wesley Ivan Hurt, Apr 06 2015

Maple

A102860:=n->8*n*(n-1)*(n-2)/3: seq(A102860(n), n=2..50); # Wesley Ivan Hurt, Apr 06 2015

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 *)

PARI

concat([0],Vec(16*x^3/(1-x)^4+O(x^40))) \\ Stefano Spezia, May 22 2021

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)