login
A081909
a(n) = 3^n(n^2 - n + 18)/18.
4
1, 3, 10, 36, 135, 513, 1944, 7290, 26973, 98415, 354294, 1259712, 4428675, 15411789, 53144100, 181752822, 617003001, 2080591515, 6973568802, 23245229340, 77096677311, 254535261273, 836828256240, 2740612539186
OFFSET
0,2
COMMENTS
Binomial transform of A081908. 3rd binomial transform of (1,0,1,0,0,0,...). Case k=3 where a(n,k) = k^n*(n^2 - n + 2k^2)/(2k^2) with g.f. (1 - 2kx + (k^2+1)x^2)/(1-kx)^3.
a(n) is the number of words of length n defined on 4 letters where one of the letters is not used or is used exactly twice. - Enrique Navarrete, Mar 29 2024
FORMULA
a(n) = 3^n*(n^2 - n + 18)/18.
G.f.: (1 - 6x + 10x^2)/(1-3x)^3.
E.g.f.: exp(3*x)*(1+x^2/2). - Enrique Navarrete, Mar 29 2024
EXAMPLE
a(2)=10 since the number of words of length 2 defined on {0,1,2,3} that don't use 0 or use it twice are 12, 21, 13, 31, 23, 32, 11, 22, 33, 00. - Enrique Navarrete, Mar 29 2024
MATHEMATICA
Table[3^n(n^2-n+18)/18, {n, 0, 30}] (* or *) CoefficientList[Series[ (1-6x+10x^2)/(1-3x)^3, {x, 0, 30}], x] (* Harvey P. Dale, Apr 26 2011 *)
PROG
(Magma) [3^n*(n^2-n+18)/18: n in [0..40]]; // Vincenzo Librandi, Apr 27 2011
CROSSREFS
Sequence in context: A371773 A272686 A126188 * A371873 A126189 A122448
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 31 2003
STATUS
approved