A129530 a(n) = (1/2)*n*(n-1)*3^(n-1).

0, 0, 3, 27, 162, 810, 3645, 15309, 61236, 236196, 885735, 3247695, 11691702, 41452398, 145083393, 502211745, 1721868840, 5854354056, 19758444939, 66248903619, 220829678730, 732224724210, 2416341589893, 7939408081077

Offset: 0

Author

Emeric Deutsch, Apr 22 2007

Keywords

• nonn
• easy

Comments

Number of inversions in all ternary words of length n on {0,1,2}. Example: a(2)=3 because each of the words 10,20,21 has one inversion and the words 00,01,02,11,12,22 have no inversions. a(n)=3*A027472(n+1). a(n)=Sum(k*A129529(n,k),k>=0).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Crossrefs

Cf. A027472, A129529, A001788, A129532.

Programs

Maple

seq(n*(n-1)*3^(n-1)/2,n=0..27);

Mathematica

Table[(n(n-1)3^(n-1))/2,{n,0,30}] (* or *) LinearRecurrence[{9,-27,27},{0,0,3},30] (* Harvey P. Dale, Dec 18 2013 *)

PARI

a(n)=n*(n-1)*3^(n-1)/2 \\ Charles R Greathouse IV, Oct 16 2015

Formula

G.f.: 3x^2/(1-3x)^3.

a(0)=0, a(1)=0, a(2)=3, a(n)=9*a(n-1)-27*a(n-2)+27*a(n-3). - Harvey P. Dale, Dec 18 2013

From Amiram Eldar, Jan 12 2021: (Start)

Sum_{n>=2} 1/a(n) = 2 * (1 - 2 * log(3/2)).

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

a(n) = 3*A027472(n+1). - R. J. Mathar, Jul 26 2022