A036290 a(n) = n*3^n.

0, 3, 18, 81, 324, 1215, 4374, 15309, 52488, 177147, 590490, 1948617, 6377292, 20726199, 66961566, 215233605, 688747536, 2195382771, 6973568802, 22082967873, 69735688020, 219667417263, 690383311398, 2165293113021, 6778308875544, 21182215236075, 66088511536554, 205891132094649

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000244, A006234, A016578, A027471, A083679, A289399 (partial sums).

Programs

Magma

[n*3^n: n in [0..30]]; // Vincenzo Librandi, Jul 06 2017

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 *)

PARI

a(n)=3^n*n \\ Charles R Greathouse IV, Jun 18 2011

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)