OFFSET
0,2
COMMENTS
If X_1,X_2,...,X_n is a partition of a 3n-set X into 3-blocks then, for n > 0, a(n) is equal to the number of (n+1)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions.
László Németh, The trinomial transform triangle, Journal of Integer Sequences, Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (6,-9).
FORMULA
From Paul Barry, Feb 06 2004: (Start)
A trinomial transform. Differentiate (1+x+x^2)^n and set x=1.
a(n) = Sum_{i=0..n} Sum_{j=0..n} (2*n-2*i-j)*n!/(i!*j!*(n-i-j)!). (End)
From Paul Barry, Feb 15 2005: (Start)
a(n) = Sum_{k=0..2*n} T(n, k)*k, where T(n, k) is given by A027907.
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k). (End)
From R. J. Mathar, Jun 19 2011: (Start)
G.f.: 3*x/(3*x-1)^2.
a(n) = 3*A027471(n+1). (End)
Sum_{n>=1} 1/a(n) = log(3/2) = 0.405465108... = A016578. - Franz Vrabec, Jan 07 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = log(4/3) = A083679. - Amiram Eldar, Jul 20 2020
a(n) = 6*a(n-1) - 9*a(n-2). - Wesley Ivan Hurt, Apr 26 2021
From Elmo R. Oliveira, Sep 09 2024: (Start)
E.g.f.: 3*x*exp(3*x).
a(n) = n*A000244(n). (End)
MAPLE
A036290 := proc(n) n*3^n ; end proc: # R. J. Mathar, Jun 18 2011
MATHEMATICA
nn=20; a=1/(1-3x); CoefficientList[Series[x D[ a, x] , {x, 0, nn}], x] (* Geoffrey Critzer, Nov 18 2012 *)
Table[n 3^n, {n, 0, 30}] (* Vincenzo Librandi, Jul 06 2017 *)
PROG
(PARI) a(n)=3^n*n \\ Charles R Greathouse IV, Jun 18, 2011
(Magma) [n*3^n: n in [0..30]]; // Vincenzo Librandi, Jul 06 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved